Weil polynomials of abelian varieties over prime fields

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Polynomials

$g$

$q$

label

$P_A(t)$

1.2.a:

t^2 + 2

comment: LMFDB 1.2.a. The class is simple, has p-rank 0, and is not ordinary.

1.2.b:

t^2 + t + 2

comment: LMFDB 1.2.b. The class is simple, has p-rank 1, and is ordinary.

1.2.ab:

t^2 - t + 2

comment: LMFDB 1.2.ab. The class is simple, has p-rank 1, and is ordinary.

1.2.c:

t^2 + 2*t + 2

comment: LMFDB 1.2.c. The class is simple, has p-rank 0, and is not ordinary.

1.2.ac:

t^2 - 2*t + 2

comment: LMFDB 1.2.ac. The class is simple, has p-rank 0, and is not ordinary.

1.3.a:

t^2 + 3

comment: LMFDB 1.3.a. The class is simple, has p-rank 0, and is not ordinary.

1.3.b:

t^2 + t + 3

comment: LMFDB 1.3.b. The class is simple, has p-rank 1, and is ordinary.

1.3.ab:

t^2 - t + 3

comment: LMFDB 1.3.ab. The class is simple, has p-rank 1, and is ordinary.

1.3.c:

t^2 + 2*t + 3

comment: LMFDB 1.3.c. The class is simple, has p-rank 1, and is ordinary.

1.3.ac:

t^2 - 2*t + 3

comment: LMFDB 1.3.ac. The class is simple, has p-rank 1, and is ordinary.

1.3.d:

t^2 + 3*t + 3

comment: LMFDB 1.3.d. The class is simple, has p-rank 0, and is not ordinary.

1.3.ad:

t^2 - 3*t + 3

comment: LMFDB 1.3.ad. The class is simple, has p-rank 0, and is not ordinary.

1.5.a:

t^2 + 5

comment: LMFDB 1.5.a. The class is simple, has p-rank 0, and is not ordinary.

1.5.b:

t^2 + t + 5

comment: LMFDB 1.5.b. The class is simple, has p-rank 1, and is ordinary.

1.5.ab:

t^2 - t + 5

comment: LMFDB 1.5.ab. The class is simple, has p-rank 1, and is ordinary.

1.5.c:

t^2 + 2*t + 5

comment: LMFDB 1.5.c. The class is simple, has p-rank 1, and is ordinary.

1.5.ac:

t^2 - 2*t + 5

comment: LMFDB 1.5.ac. The class is simple, has p-rank 1, and is ordinary.

1.5.d:

t^2 + 3*t + 5

comment: LMFDB 1.5.d. The class is simple, has p-rank 1, and is ordinary.

1.5.ad:

t^2 - 3*t + 5

comment: LMFDB 1.5.ad. The class is simple, has p-rank 1, and is ordinary.

1.5.e:

t^2 + 4*t + 5

comment: LMFDB 1.5.e. The class is simple, has p-rank 1, and is ordinary.

1.5.ae:

t^2 - 4*t + 5

comment: LMFDB 1.5.ae. The class is simple, has p-rank 1, and is ordinary.

1.7.a:

t^2 + 7

comment: LMFDB 1.7.a. The class is simple, has p-rank 0, and is not ordinary.

1.7.b:

t^2 + t + 7

comment: LMFDB 1.7.b. The class is simple, has p-rank 1, and is ordinary.

1.7.ab:

t^2 - t + 7

comment: LMFDB 1.7.ab. The class is simple, has p-rank 1, and is ordinary.

1.7.c:

t^2 + 2*t + 7

comment: LMFDB 1.7.c. The class is simple, has p-rank 1, and is ordinary.

1.7.ac:

t^2 - 2*t + 7

comment: LMFDB 1.7.ac. The class is simple, has p-rank 1, and is ordinary.

1.7.d:

t^2 + 3*t + 7

comment: LMFDB 1.7.d. The class is simple, has p-rank 1, and is ordinary.

1.7.ad:

t^2 - 3*t + 7

comment: LMFDB 1.7.ad. The class is simple, has p-rank 1, and is ordinary.

1.7.e:

t^2 + 4*t + 7

comment: LMFDB 1.7.e. The class is simple, has p-rank 1, and is ordinary.

1.7.ae:

t^2 - 4*t + 7

comment: LMFDB 1.7.ae. The class is simple, has p-rank 1, and is ordinary.

1.7.f:

t^2 + 5*t + 7

comment: LMFDB 1.7.f. The class is simple, has p-rank 1, and is ordinary.

1.7.af:

t^2 - 5*t + 7

comment: LMFDB 1.7.af. The class is simple, has p-rank 1, and is ordinary.

1.11.a:

t^2 + 11

comment: LMFDB 1.11.a. The class is simple, has p-rank 0, and is not ordinary.

1.11.b:

t^2 + t + 11

comment: LMFDB 1.11.b. The class is simple, has p-rank 1, and is ordinary.

1.11.ab:

t^2 - t + 11

comment: LMFDB 1.11.ab. The class is simple, has p-rank 1, and is ordinary.

1.11.c:

t^2 + 2*t + 11

comment: LMFDB 1.11.c. The class is simple, has p-rank 1, and is ordinary.

1.11.ac:

t^2 - 2*t + 11

comment: LMFDB 1.11.ac. The class is simple, has p-rank 1, and is ordinary.

1.11.d:

t^2 + 3*t + 11

comment: LMFDB 1.11.d. The class is simple, has p-rank 1, and is ordinary.

1.11.ad:

t^2 - 3*t + 11

comment: LMFDB 1.11.ad. The class is simple, has p-rank 1, and is ordinary.

1.11.e:

t^2 + 4*t + 11

comment: LMFDB 1.11.e. The class is simple, has p-rank 1, and is ordinary.

1.11.ae:

t^2 - 4*t + 11

comment: LMFDB 1.11.ae. The class is simple, has p-rank 1, and is ordinary.

1.11.f:

t^2 + 5*t + 11

comment: LMFDB 1.11.f. The class is simple, has p-rank 1, and is ordinary.

1.11.af:

t^2 - 5*t + 11

comment: LMFDB 1.11.af. The class is simple, has p-rank 1, and is ordinary.

1.11.g:

t^2 + 6*t + 11

comment: LMFDB 1.11.g. The class is simple, has p-rank 1, and is ordinary.

1.11.ag:

t^2 - 6*t + 11

comment: LMFDB 1.11.ag. The class is simple, has p-rank 1, and is ordinary.

1.13.a:

t^2 + 13

comment: LMFDB 1.13.a. The class is simple, has p-rank 0, and is not ordinary.

1.13.b:

t^2 + t + 13

comment: LMFDB 1.13.b. The class is simple, has p-rank 1, and is ordinary.

1.13.ab:

t^2 - t + 13

comment: LMFDB 1.13.ab. The class is simple, has p-rank 1, and is ordinary.

1.13.c:

t^2 + 2*t + 13

comment: LMFDB 1.13.c. The class is simple, has p-rank 1, and is ordinary.

1.13.ac:

t^2 - 2*t + 13

comment: LMFDB 1.13.ac. The class is simple, has p-rank 1, and is ordinary.

1.13.d:

t^2 + 3*t + 13

comment: LMFDB 1.13.d. The class is simple, has p-rank 1, and is ordinary.

1.13.ad:

t^2 - 3*t + 13

comment: LMFDB 1.13.ad. The class is simple, has p-rank 1, and is ordinary.

1.13.e:

t^2 + 4*t + 13

comment: LMFDB 1.13.e. The class is simple, has p-rank 1, and is ordinary.

1.13.ae:

t^2 - 4*t + 13

comment: LMFDB 1.13.ae. The class is simple, has p-rank 1, and is ordinary.

1.13.f:

t^2 + 5*t + 13

comment: LMFDB 1.13.f. The class is simple, has p-rank 1, and is ordinary.

1.13.af:

t^2 - 5*t + 13

comment: LMFDB 1.13.af. The class is simple, has p-rank 1, and is ordinary.

1.13.g:

t^2 + 6*t + 13

comment: LMFDB 1.13.g. The class is simple, has p-rank 1, and is ordinary.

1.13.ag:

t^2 - 6*t + 13

comment: LMFDB 1.13.ag. The class is simple, has p-rank 1, and is ordinary.

1.13.h:

t^2 + 7*t + 13

comment: LMFDB 1.13.h. The class is simple, has p-rank 1, and is ordinary.

1.13.ah:

t^2 - 7*t + 13

comment: LMFDB 1.13.ah. The class is simple, has p-rank 1, and is ordinary.

2.2.a_a:

t^4 + 4

comment: LMFDB 2.2.a_a. The class is not simple, has p-rank 0, and is not ordinary.

2.2.a_b:

t^4 + t^2 + 4

comment: LMFDB 2.2.a_b. The class is simple, has p-rank 2, and is ordinary.

2.2.a_ab:

t^4 - t^2 + 4

comment: LMFDB 2.2.a_ab. The class is simple, has p-rank 2, and is ordinary.

2.2.a_c:

t^4 + 2*t^2 + 4

comment: LMFDB 2.2.a_c. The class is simple, has p-rank 0, and is not ordinary.

2.2.a_ac:

t^4 - 2*t^2 + 4

comment: LMFDB 2.2.a_ac. The class is simple, has p-rank 0, and is not ordinary.

2.2.a_d:

t^4 + 3*t^2 + 4

comment: LMFDB 2.2.a_d. The class is not simple, has p-rank 2, and is ordinary.

2.2.a_ad:

t^4 - 3*t^2 + 4

comment: LMFDB 2.2.a_ad. The class is simple, has p-rank 2, and is ordinary.

2.2.a_e:

t^4 + 4*t^2 + 4

comment: LMFDB 2.2.a_e. The class is not simple, has p-rank 0, and is not ordinary.

2.2.a_ae:

t^4 - 4*t^2 + 4

comment: LMFDB 2.2.a_ae. The class is simple, has p-rank 0, and is not ordinary.

2.2.b_a:

t^4 + t^3 + 2*t + 4

comment: LMFDB 2.2.b_a. The class is simple, has p-rank 1, and is not ordinary.

2.2.b_b:

t^4 + t^3 + t^2 + 2*t + 4

comment: LMFDB 2.2.b_b. The class is simple, has p-rank 2, and is ordinary.

2.2.b_ab:

t^4 + t^3 - t^2 + 2*t + 4

comment: LMFDB 2.2.b_ab. The class is simple, has p-rank 2, and is ordinary.

2.2.b_c:

t^4 + t^3 + 2*t^2 + 2*t + 4

comment: LMFDB 2.2.b_c. The class is not simple, has p-rank 1, and is not ordinary.

2.2.b_d:

t^4 + t^3 + 3*t^2 + 2*t + 4

comment: LMFDB 2.2.b_d. The class is simple, has p-rank 2, and is ordinary.

2.2.b_e:

t^4 + t^3 + 4*t^2 + 2*t + 4

comment: LMFDB 2.2.b_e. The class is not simple, has p-rank 1, and is not ordinary.

2.2.ab_a:

t^4 - t^3 - 2*t + 4

comment: LMFDB 2.2.ab_a. The class is simple, has p-rank 1, and is not ordinary.

2.2.ab_b:

t^4 - t^3 + t^2 - 2*t + 4

comment: LMFDB 2.2.ab_b. The class is simple, has p-rank 2, and is ordinary.

2.2.ab_ab:

t^4 - t^3 - t^2 - 2*t + 4

comment: LMFDB 2.2.ab_ab. The class is simple, has p-rank 2, and is ordinary.

2.2.ab_c:

t^4 - t^3 + 2*t^2 - 2*t + 4

comment: LMFDB 2.2.ab_c. The class is not simple, has p-rank 1, and is not ordinary.

2.2.ab_d:

t^4 - t^3 + 3*t^2 - 2*t + 4

comment: LMFDB 2.2.ab_d. The class is simple, has p-rank 2, and is ordinary.

2.2.ab_e:

t^4 - t^3 + 4*t^2 - 2*t + 4

comment: LMFDB 2.2.ab_e. The class is not simple, has p-rank 1, and is not ordinary.

2.2.c_c:

t^4 + 2*t^3 + 2*t^2 + 4*t + 4

comment: LMFDB 2.2.c_c. The class is simple, has p-rank 0, and is not ordinary.

2.2.c_d:

t^4 + 2*t^3 + 3*t^2 + 4*t + 4

comment: LMFDB 2.2.c_d. The class is simple, has p-rank 2, and is ordinary.

2.2.c_e:

t^4 + 2*t^3 + 4*t^2 + 4*t + 4

comment: LMFDB 2.2.c_e. The class is not simple, has p-rank 0, and is not ordinary.

2.2.c_f:

t^4 + 2*t^3 + 5*t^2 + 4*t + 4

comment: LMFDB 2.2.c_f. The class is not simple, has p-rank 2, and is ordinary.

2.2.ac_c:

t^4 - 2*t^3 + 2*t^2 - 4*t + 4

comment: LMFDB 2.2.ac_c. The class is simple, has p-rank 0, and is not ordinary.

2.2.ac_d:

t^4 - 2*t^3 + 3*t^2 - 4*t + 4

comment: LMFDB 2.2.ac_d. The class is simple, has p-rank 2, and is ordinary.

2.2.ac_e:

t^4 - 2*t^3 + 4*t^2 - 4*t + 4

comment: LMFDB 2.2.ac_e. The class is not simple, has p-rank 0, and is not ordinary.

2.2.ac_f:

t^4 - 2*t^3 + 5*t^2 - 4*t + 4

comment: LMFDB 2.2.ac_f. The class is not simple, has p-rank 2, and is ordinary.

2.2.d_f:

t^4 + 3*t^3 + 5*t^2 + 6*t + 4

comment: LMFDB 2.2.d_f. The class is simple, has p-rank 2, and is ordinary.

2.2.d_g:

t^4 + 3*t^3 + 6*t^2 + 6*t + 4

comment: LMFDB 2.2.d_g. The class is not simple, has p-rank 1, and is not ordinary.

2.2.ad_f:

t^4 - 3*t^3 + 5*t^2 - 6*t + 4

comment: LMFDB 2.2.ad_f. The class is simple, has p-rank 2, and is ordinary.

2.2.ad_g:

t^4 - 3*t^3 + 6*t^2 - 6*t + 4

comment: LMFDB 2.2.ad_g. The class is not simple, has p-rank 1, and is not ordinary.

2.2.e_i:

t^4 + 4*t^3 + 8*t^2 + 8*t + 4

comment: LMFDB 2.2.e_i. The class is not simple, has p-rank 0, and is not ordinary.

2.2.ae_i:

t^4 - 4*t^3 + 8*t^2 - 8*t + 4

comment: LMFDB 2.2.ae_i. The class is not simple, has p-rank 0, and is not ordinary.

2.3.a_a:

t^4 + 9

comment: LMFDB 2.3.a_a. The class is simple, has p-rank 0, and is not ordinary.

2.3.a_b:

t^4 + t^2 + 9

comment: LMFDB 2.3.a_b. The class is simple, has p-rank 2, and is ordinary.

2.3.a_ab:

t^4 - t^2 + 9

comment: LMFDB 2.3.a_ab. The class is simple, has p-rank 2, and is ordinary.

2.3.a_c:

t^4 + 2*t^2 + 9

comment: LMFDB 2.3.a_c. The class is not simple, has p-rank 2, and is ordinary.

2.3.a_ac:

t^4 - 2*t^2 + 9

comment: LMFDB 2.3.a_ac. The class is simple, has p-rank 2, and is ordinary.

2.3.a_d:

t^4 + 3*t^2 + 9

comment: LMFDB 2.3.a_d. The class is simple, has p-rank 0, and is not ordinary.

2.3.a_ad:

t^4 - 3*t^2 + 9

comment: LMFDB 2.3.a_ad. The class is not simple, has p-rank 0, and is not ordinary.

2.3.a_e:

t^4 + 4*t^2 + 9

comment: LMFDB 2.3.a_e. The class is simple, has p-rank 2, and is ordinary.

2.3.a_ae:

t^4 - 4*t^2 + 9

comment: LMFDB 2.3.a_ae. The class is simple, has p-rank 2, and is ordinary.

2.3.a_f:

t^4 + 5*t^2 + 9

comment: LMFDB 2.3.a_f. The class is not simple, has p-rank 2, and is ordinary.

2.3.a_af:

t^4 - 5*t^2 + 9

comment: LMFDB 2.3.a_af. The class is simple, has p-rank 2, and is ordinary.

2.3.a_g:

t^4 + 6*t^2 + 9

comment: LMFDB 2.3.a_g. The class is not simple, has p-rank 0, and is not ordinary.

2.3.a_ag:

t^4 - 6*t^2 + 9

comment: LMFDB 2.3.a_ag. The class is simple, has p-rank 0, and is not ordinary.

2.3.b_a:

t^4 + t^3 + 3*t + 9

comment: LMFDB 2.3.b_a. The class is not simple, has p-rank 1, and is not ordinary.

2.3.b_b:

t^4 + t^3 + t^2 + 3*t + 9

comment: LMFDB 2.3.b_b. The class is simple, has p-rank 2, and is ordinary.

2.3.b_ab:

t^4 + t^3 - t^2 + 3*t + 9

comment: LMFDB 2.3.b_ab. The class is simple, has p-rank 2, and is ordinary.

2.3.b_c:

t^4 + t^3 + 2*t^2 + 3*t + 9

comment: LMFDB 2.3.b_c. The class is simple, has p-rank 2, and is ordinary.

2.3.b_ac:

t^4 + t^3 - 2*t^2 + 3*t + 9

comment: LMFDB 2.3.b_ac. The class is simple, has p-rank 2, and is ordinary.

2.3.b_d:

t^4 + t^3 + 3*t^2 + 3*t + 9

comment: LMFDB 2.3.b_d. The class is simple, has p-rank 1, and is not ordinary.

2.3.b_e:

t^4 + t^3 + 4*t^2 + 3*t + 9

comment: LMFDB 2.3.b_e. The class is not simple, has p-rank 2, and is ordinary.

2.3.b_f:

t^4 + t^3 + 5*t^2 + 3*t + 9

comment: LMFDB 2.3.b_f. The class is simple, has p-rank 2, and is ordinary.

2.3.b_g:

t^4 + t^3 + 6*t^2 + 3*t + 9

comment: LMFDB 2.3.b_g. The class is not simple, has p-rank 1, and is not ordinary.

2.3.ab_a:

t^4 - t^3 - 3*t + 9

comment: LMFDB 2.3.ab_a. The class is not simple, has p-rank 1, and is not ordinary.

2.3.ab_b:

t^4 - t^3 + t^2 - 3*t + 9

comment: LMFDB 2.3.ab_b. The class is simple, has p-rank 2, and is ordinary.

2.3.ab_ab:

t^4 - t^3 - t^2 - 3*t + 9

comment: LMFDB 2.3.ab_ab. The class is simple, has p-rank 2, and is ordinary.

2.3.ab_c:

t^4 - t^3 + 2*t^2 - 3*t + 9

comment: LMFDB 2.3.ab_c. The class is simple, has p-rank 2, and is ordinary.

2.3.ab_ac:

t^4 - t^3 - 2*t^2 - 3*t + 9

comment: LMFDB 2.3.ab_ac. The class is simple, has p-rank 2, and is ordinary.

2.3.ab_d:

t^4 - t^3 + 3*t^2 - 3*t + 9

comment: LMFDB 2.3.ab_d. The class is simple, has p-rank 1, and is not ordinary.

2.3.ab_e:

t^4 - t^3 + 4*t^2 - 3*t + 9

comment: LMFDB 2.3.ab_e. The class is not simple, has p-rank 2, and is ordinary.

2.3.ab_f:

t^4 - t^3 + 5*t^2 - 3*t + 9

comment: LMFDB 2.3.ab_f. The class is simple, has p-rank 2, and is ordinary.

2.3.ab_g:

t^4 - t^3 + 6*t^2 - 3*t + 9

comment: LMFDB 2.3.ab_g. The class is not simple, has p-rank 1, and is not ordinary.

2.3.c_b:

t^4 + 2*t^3 + t^2 + 6*t + 9

comment: LMFDB 2.3.c_b. The class is simple, has p-rank 2, and is ordinary.

2.3.c_c:

t^4 + 2*t^3 + 2*t^2 + 6*t + 9

comment: LMFDB 2.3.c_c. The class is simple, has p-rank 2, and is ordinary.

2.3.c_d:

t^4 + 2*t^3 + 3*t^2 + 6*t + 9

comment: LMFDB 2.3.c_d. The class is not simple, has p-rank 1, and is not ordinary.

2.3.c_e:

t^4 + 2*t^3 + 4*t^2 + 6*t + 9

comment: LMFDB 2.3.c_e. The class is simple, has p-rank 2, and is ordinary.

2.3.c_f:

t^4 + 2*t^3 + 5*t^2 + 6*t + 9

comment: LMFDB 2.3.c_f. The class is simple, has p-rank 2, and is ordinary.

2.3.c_g:

t^4 + 2*t^3 + 6*t^2 + 6*t + 9

comment: LMFDB 2.3.c_g. The class is not simple, has p-rank 1, and is not ordinary.

2.3.c_h:

t^4 + 2*t^3 + 7*t^2 + 6*t + 9

comment: LMFDB 2.3.c_h. The class is not simple, has p-rank 2, and is ordinary.

2.3.ac_b:

t^4 - 2*t^3 + t^2 - 6*t + 9

comment: LMFDB 2.3.ac_b. The class is simple, has p-rank 2, and is ordinary.

2.3.ac_c:

t^4 - 2*t^3 + 2*t^2 - 6*t + 9

comment: LMFDB 2.3.ac_c. The class is simple, has p-rank 2, and is ordinary.

2.3.ac_d:

t^4 - 2*t^3 + 3*t^2 - 6*t + 9

comment: LMFDB 2.3.ac_d. The class is not simple, has p-rank 1, and is not ordinary.

2.3.ac_e:

t^4 - 2*t^3 + 4*t^2 - 6*t + 9

comment: LMFDB 2.3.ac_e. The class is simple, has p-rank 2, and is ordinary.

2.3.ac_f:

t^4 - 2*t^3 + 5*t^2 - 6*t + 9

comment: LMFDB 2.3.ac_f. The class is simple, has p-rank 2, and is ordinary.

2.3.ac_g:

t^4 - 2*t^3 + 6*t^2 - 6*t + 9

comment: LMFDB 2.3.ac_g. The class is not simple, has p-rank 1, and is not ordinary.

2.3.ac_h:

t^4 - 2*t^3 + 7*t^2 - 6*t + 9

comment: LMFDB 2.3.ac_h. The class is not simple, has p-rank 2, and is ordinary.

2.3.d_f:

t^4 + 3*t^3 + 5*t^2 + 9*t + 9

comment: LMFDB 2.3.d_f. The class is simple, has p-rank 2, and is ordinary.

2.3.d_g:

t^4 + 3*t^3 + 6*t^2 + 9*t + 9

comment: LMFDB 2.3.d_g. The class is not simple, has p-rank 0, and is not ordinary.

2.3.d_h:

t^4 + 3*t^3 + 7*t^2 + 9*t + 9

comment: LMFDB 2.3.d_h. The class is simple, has p-rank 2, and is ordinary.

2.3.d_i:

t^4 + 3*t^3 + 8*t^2 + 9*t + 9

comment: LMFDB 2.3.d_i. The class is not simple, has p-rank 2, and is ordinary.

2.3.ad_f:

t^4 - 3*t^3 + 5*t^2 - 9*t + 9

comment: LMFDB 2.3.ad_f. The class is simple, has p-rank 2, and is ordinary.

2.3.ad_g:

t^4 - 3*t^3 + 6*t^2 - 9*t + 9

comment: LMFDB 2.3.ad_g. The class is not simple, has p-rank 0, and is not ordinary.

2.3.ad_h:

t^4 - 3*t^3 + 7*t^2 - 9*t + 9

comment: LMFDB 2.3.ad_h. The class is simple, has p-rank 2, and is ordinary.

2.3.ad_i:

t^4 - 3*t^3 + 8*t^2 - 9*t + 9

comment: LMFDB 2.3.ad_i. The class is not simple, has p-rank 2, and is ordinary.

2.3.e_i:

t^4 + 4*t^3 + 8*t^2 + 12*t + 9

comment: LMFDB 2.3.e_i. The class is simple, has p-rank 2, and is ordinary.

2.3.e_j:

t^4 + 4*t^3 + 9*t^2 + 12*t + 9

comment: LMFDB 2.3.e_j. The class is not simple, has p-rank 1, and is not ordinary.

2.3.e_k:

t^4 + 4*t^3 + 10*t^2 + 12*t + 9

comment: LMFDB 2.3.e_k. The class is not simple, has p-rank 2, and is ordinary.

2.3.ae_i:

t^4 - 4*t^3 + 8*t^2 - 12*t + 9

comment: LMFDB 2.3.ae_i. The class is simple, has p-rank 2, and is ordinary.

2.3.ae_j:

t^4 - 4*t^3 + 9*t^2 - 12*t + 9

comment: LMFDB 2.3.ae_j. The class is not simple, has p-rank 1, and is not ordinary.

2.3.ae_k:

t^4 - 4*t^3 + 10*t^2 - 12*t + 9

comment: LMFDB 2.3.ae_k. The class is not simple, has p-rank 2, and is ordinary.

2.3.f_m:

t^4 + 5*t^3 + 12*t^2 + 15*t + 9

comment: LMFDB 2.3.f_m. The class is not simple, has p-rank 1, and is not ordinary.

2.3.af_m:

t^4 - 5*t^3 + 12*t^2 - 15*t + 9

comment: LMFDB 2.3.af_m. The class is not simple, has p-rank 1, and is not ordinary.

2.3.g_p:

t^4 + 6*t^3 + 15*t^2 + 18*t + 9

comment: LMFDB 2.3.g_p. The class is not simple, has p-rank 0, and is not ordinary.

2.3.ag_p:

t^4 - 6*t^3 + 15*t^2 - 18*t + 9

comment: LMFDB 2.3.ag_p. The class is not simple, has p-rank 0, and is not ordinary.

2.5.a_a:

t^4 + 25

comment: LMFDB 2.5.a_a. The class is simple, has p-rank 0, and is not ordinary.

2.5.a_b:

t^4 + t^2 + 25

comment: LMFDB 2.5.a_b. The class is not simple, has p-rank 2, and is ordinary.

2.5.a_ab:

t^4 - t^2 + 25

comment: LMFDB 2.5.a_ab. The class is simple, has p-rank 2, and is ordinary.

2.5.a_c:

t^4 + 2*t^2 + 25

comment: LMFDB 2.5.a_c. The class is simple, has p-rank 2, and is ordinary.

2.5.a_ac:

t^4 - 2*t^2 + 25

comment: LMFDB 2.5.a_ac. The class is simple, has p-rank 2, and is ordinary.

2.5.a_d:

t^4 + 3*t^2 + 25

comment: LMFDB 2.5.a_d. The class is simple, has p-rank 2, and is ordinary.

2.5.a_ad:

t^4 - 3*t^2 + 25

comment: LMFDB 2.5.a_ad. The class is simple, has p-rank 2, and is ordinary.

2.5.a_e:

t^4 + 4*t^2 + 25

comment: LMFDB 2.5.a_e. The class is simple, has p-rank 2, and is ordinary.

2.5.a_ae:

t^4 - 4*t^2 + 25

comment: LMFDB 2.5.a_ae. The class is simple, has p-rank 2, and is ordinary.

2.5.a_f:

t^4 + 5*t^2 + 25

comment: LMFDB 2.5.a_f. The class is simple, has p-rank 0, and is not ordinary.

2.5.a_af:

t^4 - 5*t^2 + 25

comment: LMFDB 2.5.a_af. The class is simple, has p-rank 0, and is not ordinary.

2.5.a_g:

t^4 + 6*t^2 + 25

comment: LMFDB 2.5.a_g. The class is not simple, has p-rank 2, and is ordinary.

2.5.a_ag:

t^4 - 6*t^2 + 25

comment: LMFDB 2.5.a_ag. The class is not simple, has p-rank 2, and is ordinary.

2.5.a_h:

t^4 + 7*t^2 + 25

comment: LMFDB 2.5.a_h. The class is simple, has p-rank 2, and is ordinary.

2.5.a_ah:

t^4 - 7*t^2 + 25

comment: LMFDB 2.5.a_ah. The class is simple, has p-rank 2, and is ordinary.

2.5.a_i:

t^4 + 8*t^2 + 25

comment: LMFDB 2.5.a_i. The class is simple, has p-rank 2, and is ordinary.

2.5.a_ai:

t^4 - 8*t^2 + 25

comment: LMFDB 2.5.a_ai. The class is simple, has p-rank 2, and is ordinary.

2.5.a_j:

t^4 + 9*t^2 + 25

comment: LMFDB 2.5.a_j. The class is not simple, has p-rank 2, and is ordinary.

2.5.a_aj:

t^4 - 9*t^2 + 25

comment: LMFDB 2.5.a_aj. The class is simple, has p-rank 2, and is ordinary.

2.5.a_k:

t^4 + 10*t^2 + 25

comment: LMFDB 2.5.a_k. The class is not simple, has p-rank 0, and is not ordinary.

2.5.a_ak:

t^4 - 10*t^2 + 25

comment: LMFDB 2.5.a_ak. The class is simple, has p-rank 0, and is not ordinary.

2.5.b_a:

t^4 + t^3 + 5*t + 25

comment: LMFDB 2.5.b_a. The class is simple, has p-rank 1, and is not ordinary.

2.5.b_b:

t^4 + t^3 + t^2 + 5*t + 25

comment: LMFDB 2.5.b_b. The class is simple, has p-rank 2, and is ordinary.

2.5.b_ab:

t^4 + t^3 - t^2 + 5*t + 25

comment: LMFDB 2.5.b_ab. The class is simple, has p-rank 2, and is ordinary.

2.5.b_c:

t^4 + t^3 + 2*t^2 + 5*t + 25

comment: LMFDB 2.5.b_c. The class is simple, has p-rank 2, and is ordinary.

2.5.b_ac:

t^4 + t^3 - 2*t^2 + 5*t + 25

comment: LMFDB 2.5.b_ac. The class is not simple, has p-rank 2, and is ordinary.

2.5.b_d:

t^4 + t^3 + 3*t^2 + 5*t + 25

comment: LMFDB 2.5.b_d. The class is simple, has p-rank 2, and is ordinary.

2.5.b_ad:

t^4 + t^3 - 3*t^2 + 5*t + 25

comment: LMFDB 2.5.b_ad. The class is simple, has p-rank 2, and is ordinary.

2.5.b_e:

t^4 + t^3 + 4*t^2 + 5*t + 25

comment: LMFDB 2.5.b_e. The class is not simple, has p-rank 2, and is ordinary.

2.5.b_ae:

t^4 + t^3 - 4*t^2 + 5*t + 25

comment: LMFDB 2.5.b_ae. The class is simple, has p-rank 2, and is ordinary.

2.5.b_f:

t^4 + t^3 + 5*t^2 + 5*t + 25

comment: LMFDB 2.5.b_f. The class is simple, has p-rank 1, and is not ordinary.

2.5.b_af:

t^4 + t^3 - 5*t^2 + 5*t + 25

comment: LMFDB 2.5.b_af. The class is simple, has p-rank 1, and is not ordinary.

2.5.b_g:

t^4 + t^3 + 6*t^2 + 5*t + 25

comment: LMFDB 2.5.b_g. The class is simple, has p-rank 2, and is ordinary.

2.5.b_h:

t^4 + t^3 + 7*t^2 + 5*t + 25

comment: LMFDB 2.5.b_h. The class is simple, has p-rank 2, and is ordinary.

2.5.b_i:

t^4 + t^3 + 8*t^2 + 5*t + 25

comment: LMFDB 2.5.b_i. The class is not simple, has p-rank 2, and is ordinary.

2.5.b_j:

t^4 + t^3 + 9*t^2 + 5*t + 25

comment: LMFDB 2.5.b_j. The class is simple, has p-rank 2, and is ordinary.

2.5.b_k:

t^4 + t^3 + 10*t^2 + 5*t + 25

comment: LMFDB 2.5.b_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.ab_a:

t^4 - t^3 - 5*t + 25

comment: LMFDB 2.5.ab_a. The class is simple, has p-rank 1, and is not ordinary.

2.5.ab_b:

t^4 - t^3 + t^2 - 5*t + 25

comment: LMFDB 2.5.ab_b. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_ab:

t^4 - t^3 - t^2 - 5*t + 25

comment: LMFDB 2.5.ab_ab. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_c:

t^4 - t^3 + 2*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_c. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_ac:

t^4 - t^3 - 2*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_ac. The class is not simple, has p-rank 2, and is ordinary.

2.5.ab_d:

t^4 - t^3 + 3*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_d. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_ad:

t^4 - t^3 - 3*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_ad. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_e:

t^4 - t^3 + 4*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_e. The class is not simple, has p-rank 2, and is ordinary.

2.5.ab_ae:

t^4 - t^3 - 4*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_ae. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_f:

t^4 - t^3 + 5*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_f. The class is simple, has p-rank 1, and is not ordinary.

2.5.ab_af:

t^4 - t^3 - 5*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_af. The class is simple, has p-rank 1, and is not ordinary.

2.5.ab_g:

t^4 - t^3 + 6*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_g. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_h:

t^4 - t^3 + 7*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_h. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_i:

t^4 - t^3 + 8*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_i. The class is not simple, has p-rank 2, and is ordinary.

2.5.ab_j:

t^4 - t^3 + 9*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_j. The class is simple, has p-rank 2, and is ordinary.

2.5.ab_k:

t^4 - t^3 + 10*t^2 - 5*t + 25

comment: LMFDB 2.5.ab_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.c_a:

t^4 + 2*t^3 + 10*t + 25

comment: LMFDB 2.5.c_a. The class is simple, has p-rank 1, and is not ordinary.

2.5.c_b:

t^4 + 2*t^3 + t^2 + 10*t + 25

comment: LMFDB 2.5.c_b. The class is simple, has p-rank 2, and is ordinary.

2.5.c_ab:

t^4 + 2*t^3 - t^2 + 10*t + 25

comment: LMFDB 2.5.c_ab. The class is simple, has p-rank 2, and is ordinary.

2.5.c_c:

t^4 + 2*t^3 + 2*t^2 + 10*t + 25

comment: LMFDB 2.5.c_c. The class is not simple, has p-rank 2, and is ordinary.

2.5.c_d:

t^4 + 2*t^3 + 3*t^2 + 10*t + 25

comment: LMFDB 2.5.c_d. The class is simple, has p-rank 2, and is ordinary.

2.5.c_e:

t^4 + 2*t^3 + 4*t^2 + 10*t + 25

comment: LMFDB 2.5.c_e. The class is simple, has p-rank 2, and is ordinary.

2.5.c_f:

t^4 + 2*t^3 + 5*t^2 + 10*t + 25

comment: LMFDB 2.5.c_f. The class is simple, has p-rank 1, and is not ordinary.

2.5.c_g:

t^4 + 2*t^3 + 6*t^2 + 10*t + 25

comment: LMFDB 2.5.c_g. The class is simple, has p-rank 2, and is ordinary.

2.5.c_h:

t^4 + 2*t^3 + 7*t^2 + 10*t + 25

comment: LMFDB 2.5.c_h. The class is not simple, has p-rank 2, and is ordinary.

2.5.c_i:

t^4 + 2*t^3 + 8*t^2 + 10*t + 25

comment: LMFDB 2.5.c_i. The class is simple, has p-rank 2, and is ordinary.

2.5.c_j:

t^4 + 2*t^3 + 9*t^2 + 10*t + 25

comment: LMFDB 2.5.c_j. The class is simple, has p-rank 2, and is ordinary.

2.5.c_k:

t^4 + 2*t^3 + 10*t^2 + 10*t + 25

comment: LMFDB 2.5.c_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.c_l:

t^4 + 2*t^3 + 11*t^2 + 10*t + 25

comment: LMFDB 2.5.c_l. The class is not simple, has p-rank 2, and is ordinary.

2.5.ac_a:

t^4 - 2*t^3 - 10*t + 25

comment: LMFDB 2.5.ac_a. The class is simple, has p-rank 1, and is not ordinary.

2.5.ac_b:

t^4 - 2*t^3 + t^2 - 10*t + 25

comment: LMFDB 2.5.ac_b. The class is simple, has p-rank 2, and is ordinary.

2.5.ac_ab:

t^4 - 2*t^3 - t^2 - 10*t + 25

comment: LMFDB 2.5.ac_ab. The class is simple, has p-rank 2, and is ordinary.

2.5.ac_c:

t^4 - 2*t^3 + 2*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_c. The class is not simple, has p-rank 2, and is ordinary.

2.5.ac_d:

t^4 - 2*t^3 + 3*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_d. The class is simple, has p-rank 2, and is ordinary.

2.5.ac_e:

t^4 - 2*t^3 + 4*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_e. The class is simple, has p-rank 2, and is ordinary.

2.5.ac_f:

t^4 - 2*t^3 + 5*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_f. The class is simple, has p-rank 1, and is not ordinary.

2.5.ac_g:

t^4 - 2*t^3 + 6*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_g. The class is simple, has p-rank 2, and is ordinary.

2.5.ac_h:

t^4 - 2*t^3 + 7*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_h. The class is not simple, has p-rank 2, and is ordinary.

2.5.ac_i:

t^4 - 2*t^3 + 8*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_i. The class is simple, has p-rank 2, and is ordinary.

2.5.ac_j:

t^4 - 2*t^3 + 9*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_j. The class is simple, has p-rank 2, and is ordinary.

2.5.ac_k:

t^4 - 2*t^3 + 10*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.ac_l:

t^4 - 2*t^3 + 11*t^2 - 10*t + 25

comment: LMFDB 2.5.ac_l. The class is not simple, has p-rank 2, and is ordinary.

2.5.d_e:

t^4 + 3*t^3 + 4*t^2 + 15*t + 25

comment: LMFDB 2.5.d_e. The class is simple, has p-rank 2, and is ordinary.

2.5.d_f:

t^4 + 3*t^3 + 5*t^2 + 15*t + 25

comment: LMFDB 2.5.d_f. The class is simple, has p-rank 1, and is not ordinary.

2.5.d_g:

t^4 + 3*t^3 + 6*t^2 + 15*t + 25

comment: LMFDB 2.5.d_g. The class is not simple, has p-rank 2, and is ordinary.

2.5.d_h:

t^4 + 3*t^3 + 7*t^2 + 15*t + 25

comment: LMFDB 2.5.d_h. The class is simple, has p-rank 2, and is ordinary.

2.5.d_i:

t^4 + 3*t^3 + 8*t^2 + 15*t + 25

comment: LMFDB 2.5.d_i. The class is simple, has p-rank 2, and is ordinary.

2.5.d_j:

t^4 + 3*t^3 + 9*t^2 + 15*t + 25

comment: LMFDB 2.5.d_j. The class is simple, has p-rank 2, and is ordinary.

2.5.d_k:

t^4 + 3*t^3 + 10*t^2 + 15*t + 25

comment: LMFDB 2.5.d_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.d_l:

t^4 + 3*t^3 + 11*t^2 + 15*t + 25

comment: LMFDB 2.5.d_l. The class is simple, has p-rank 2, and is ordinary.

2.5.d_m:

t^4 + 3*t^3 + 12*t^2 + 15*t + 25

comment: LMFDB 2.5.d_m. The class is not simple, has p-rank 2, and is ordinary.

2.5.ad_e:

t^4 - 3*t^3 + 4*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_e. The class is simple, has p-rank 2, and is ordinary.

2.5.ad_f:

t^4 - 3*t^3 + 5*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_f. The class is simple, has p-rank 1, and is not ordinary.

2.5.ad_g:

t^4 - 3*t^3 + 6*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_g. The class is not simple, has p-rank 2, and is ordinary.

2.5.ad_h:

t^4 - 3*t^3 + 7*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_h. The class is simple, has p-rank 2, and is ordinary.

2.5.ad_i:

t^4 - 3*t^3 + 8*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_i. The class is simple, has p-rank 2, and is ordinary.

2.5.ad_j:

t^4 - 3*t^3 + 9*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_j. The class is simple, has p-rank 2, and is ordinary.

2.5.ad_k:

t^4 - 3*t^3 + 10*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.ad_l:

t^4 - 3*t^3 + 11*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_l. The class is simple, has p-rank 2, and is ordinary.

2.5.ad_m:

t^4 - 3*t^3 + 12*t^2 - 15*t + 25

comment: LMFDB 2.5.ad_m. The class is not simple, has p-rank 2, and is ordinary.

2.5.e_i:

t^4 + 4*t^3 + 8*t^2 + 20*t + 25

comment: LMFDB 2.5.e_i. The class is simple, has p-rank 2, and is ordinary.

2.5.e_j:

t^4 + 4*t^3 + 9*t^2 + 20*t + 25

comment: LMFDB 2.5.e_j. The class is simple, has p-rank 2, and is ordinary.

2.5.e_k:

t^4 + 4*t^3 + 10*t^2 + 20*t + 25

comment: LMFDB 2.5.e_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.e_l:

t^4 + 4*t^3 + 11*t^2 + 20*t + 25

comment: LMFDB 2.5.e_l. The class is simple, has p-rank 2, and is ordinary.

2.5.e_m:

t^4 + 4*t^3 + 12*t^2 + 20*t + 25

comment: LMFDB 2.5.e_m. The class is simple, has p-rank 2, and is ordinary.

2.5.e_n:

t^4 + 4*t^3 + 13*t^2 + 20*t + 25

comment: LMFDB 2.5.e_n. The class is not simple, has p-rank 2, and is ordinary.

2.5.e_o:

t^4 + 4*t^3 + 14*t^2 + 20*t + 25

comment: LMFDB 2.5.e_o. The class is not simple, has p-rank 2, and is ordinary.

2.5.ae_i:

t^4 - 4*t^3 + 8*t^2 - 20*t + 25

comment: LMFDB 2.5.ae_i. The class is simple, has p-rank 2, and is ordinary.

2.5.ae_j:

t^4 - 4*t^3 + 9*t^2 - 20*t + 25

comment: LMFDB 2.5.ae_j. The class is simple, has p-rank 2, and is ordinary.

2.5.ae_k:

t^4 - 4*t^3 + 10*t^2 - 20*t + 25

comment: LMFDB 2.5.ae_k. The class is not simple, has p-rank 1, and is not ordinary.

2.5.ae_l:

t^4 - 4*t^3 + 11*t^2 - 20*t + 25

comment: LMFDB 2.5.ae_l. The class is simple, has p-rank 2, and is ordinary.

2.5.ae_m:

t^4 - 4*t^3 + 12*t^2 - 20*t + 25

comment: LMFDB 2.5.ae_m. The class is simple, has p-rank 2, and is ordinary.

2.5.ae_n:

t^4 - 4*t^3 + 13*t^2 - 20*t + 25

comment: LMFDB 2.5.ae_n. The class is not simple, has p-rank 2, and is ordinary.

2.5.ae_o:

t^4 - 4*t^3 + 14*t^2 - 20*t + 25

comment: LMFDB 2.5.ae_o. The class is not simple, has p-rank 2, and is ordinary.

2.5.f_n:

t^4 + 5*t^3 + 13*t^2 + 25*t + 25

comment: LMFDB 2.5.f_n. The class is simple, has p-rank 2, and is ordinary.

2.5.f_o:

t^4 + 5*t^3 + 14*t^2 + 25*t + 25

comment: LMFDB 2.5.f_o. The class is not simple, has p-rank 2, and is ordinary.

2.5.f_p:

t^4 + 5*t^3 + 15*t^2 + 25*t + 25

comment: LMFDB 2.5.f_p. The class is simple, has p-rank 0, and is not ordinary.

2.5.f_q:

t^4 + 5*t^3 + 16*t^2 + 25*t + 25

comment: LMFDB 2.5.f_q. The class is not simple, has p-rank 2, and is ordinary.

2.5.af_n:

t^4 - 5*t^3 + 13*t^2 - 25*t + 25

comment: LMFDB 2.5.af_n. The class is simple, has p-rank 2, and is ordinary.

2.5.af_o:

t^4 - 5*t^3 + 14*t^2 - 25*t + 25

comment: LMFDB 2.5.af_o. The class is not simple, has p-rank 2, and is ordinary.

2.5.af_p:

t^4 - 5*t^3 + 15*t^2 - 25*t + 25

comment: LMFDB 2.5.af_p. The class is simple, has p-rank 0, and is not ordinary.

2.5.af_q:

t^4 - 5*t^3 + 16*t^2 - 25*t + 25

comment: LMFDB 2.5.af_q. The class is not simple, has p-rank 2, and is ordinary.

2.5.g_r:

t^4 + 6*t^3 + 17*t^2 + 30*t + 25

comment: LMFDB 2.5.g_r. The class is simple, has p-rank 2, and is ordinary.

2.5.g_s:

t^4 + 6*t^3 + 18*t^2 + 30*t + 25

comment: LMFDB 2.5.g_s. The class is not simple, has p-rank 2, and is ordinary.

2.5.g_t:

t^4 + 6*t^3 + 19*t^2 + 30*t + 25

comment: LMFDB 2.5.g_t. The class is not simple, has p-rank 2, and is ordinary.

2.5.ag_r:

t^4 - 6*t^3 + 17*t^2 - 30*t + 25

comment: LMFDB 2.5.ag_r. The class is simple, has p-rank 2, and is ordinary.

2.5.ag_s:

t^4 - 6*t^3 + 18*t^2 - 30*t + 25

comment: LMFDB 2.5.ag_s. The class is not simple, has p-rank 2, and is ordinary.

2.5.ag_t:

t^4 - 6*t^3 + 19*t^2 - 30*t + 25

comment: LMFDB 2.5.ag_t. The class is not simple, has p-rank 2, and is ordinary.

2.5.h_w:

t^4 + 7*t^3 + 22*t^2 + 35*t + 25

comment: LMFDB 2.5.h_w. The class is not simple, has p-rank 2, and is ordinary.

2.5.ah_w:

t^4 - 7*t^3 + 22*t^2 - 35*t + 25

comment: LMFDB 2.5.ah_w. The class is not simple, has p-rank 2, and is ordinary.

2.5.i_ba:

t^4 + 8*t^3 + 26*t^2 + 40*t + 25

comment: LMFDB 2.5.i_ba. The class is not simple, has p-rank 2, and is ordinary.

2.5.ai_ba:

t^4 - 8*t^3 + 26*t^2 - 40*t + 25

comment: LMFDB 2.5.ai_ba. The class is not simple, has p-rank 2, and is ordinary.

2.7.a_a:

t^4 + 49

comment: LMFDB 2.7.a_a. The class is simple, has p-rank 0, and is not ordinary.

2.7.a_b:

t^4 + t^2 + 49

comment: LMFDB 2.7.a_b. The class is simple, has p-rank 2, and is ordinary.

2.7.a_ab:

t^4 - t^2 + 49

comment: LMFDB 2.7.a_ab. The class is simple, has p-rank 2, and is ordinary.

2.7.a_c:

t^4 + 2*t^2 + 49

comment: LMFDB 2.7.a_c. The class is simple, has p-rank 2, and is ordinary.

2.7.a_ac:

t^4 - 2*t^2 + 49

comment: LMFDB 2.7.a_ac. The class is not simple, has p-rank 2, and is ordinary.

2.7.a_d:

t^4 + 3*t^2 + 49

comment: LMFDB 2.7.a_d. The class is simple, has p-rank 2, and is ordinary.

2.7.a_ad:

t^4 - 3*t^2 + 49

comment: LMFDB 2.7.a_ad. The class is simple, has p-rank 2, and is ordinary.

2.7.a_e:

t^4 + 4*t^2 + 49

comment: LMFDB 2.7.a_e. The class is simple, has p-rank 2, and is ordinary.

2.7.a_ae:

t^4 - 4*t^2 + 49

comment: LMFDB 2.7.a_ae. The class is simple, has p-rank 2, and is ordinary.

2.7.a_f:

t^4 + 5*t^2 + 49

comment: LMFDB 2.7.a_f. The class is not simple, has p-rank 2, and is ordinary.

2.7.a_af:

t^4 - 5*t^2 + 49

comment: LMFDB 2.7.a_af. The class is simple, has p-rank 2, and is ordinary.

2.7.a_g:

t^4 + 6*t^2 + 49

comment: LMFDB 2.7.a_g. The class is simple, has p-rank 2, and is ordinary.

2.7.a_ag:

t^4 - 6*t^2 + 49

comment: LMFDB 2.7.a_ag. The class is simple, has p-rank 2, and is ordinary.

2.7.a_h:

t^4 + 7*t^2 + 49

comment: LMFDB 2.7.a_h. The class is simple, has p-rank 0, and is not ordinary.

2.7.a_ah:

t^4 - 7*t^2 + 49

comment: LMFDB 2.7.a_ah. The class is simple, has p-rank 0, and is not ordinary.

2.7.a_i:

t^4 + 8*t^2 + 49

comment: LMFDB 2.7.a_i. The class is simple, has p-rank 2, and is ordinary.

2.7.a_ai:

t^4 - 8*t^2 + 49

comment: LMFDB 2.7.a_ai. The class is simple, has p-rank 2, and is ordinary.

2.7.a_j:

t^4 + 9*t^2 + 49

comment: LMFDB 2.7.a_j. The class is simple, has p-rank 2, and is ordinary.

2.7.a_aj:

t^4 - 9*t^2 + 49

comment: LMFDB 2.7.a_aj. The class is simple, has p-rank 2, and is ordinary.

2.7.a_k:

t^4 + 10*t^2 + 49

comment: LMFDB 2.7.a_k. The class is not simple, has p-rank 2, and is ordinary.

2.7.a_ak:

t^4 - 10*t^2 + 49

comment: LMFDB 2.7.a_ak. The class is simple, has p-rank 2, and is ordinary.

2.7.a_l:

t^4 + 11*t^2 + 49

comment: LMFDB 2.7.a_l. The class is simple, has p-rank 2, and is ordinary.

2.7.a_al:

t^4 - 11*t^2 + 49

comment: LMFDB 2.7.a_al. The class is not simple, has p-rank 2, and is ordinary.

2.7.a_m:

t^4 + 12*t^2 + 49

comment: LMFDB 2.7.a_m. The class is simple, has p-rank 2, and is ordinary.

2.7.a_am:

t^4 - 12*t^2 + 49

comment: LMFDB 2.7.a_am. The class is simple, has p-rank 2, and is ordinary.

2.7.a_n:

t^4 + 13*t^2 + 49

comment: LMFDB 2.7.a_n. The class is not simple, has p-rank 2, and is ordinary.

2.7.a_an:

t^4 - 13*t^2 + 49

comment: LMFDB 2.7.a_an. The class is simple, has p-rank 2, and is ordinary.

2.7.a_o:

t^4 + 14*t^2 + 49

comment: LMFDB 2.7.a_o. The class is not simple, has p-rank 0, and is not ordinary.

2.7.a_ao:

t^4 - 14*t^2 + 49

comment: LMFDB 2.7.a_ao. The class is simple, has p-rank 0, and is not ordinary.

2.7.b_a:

t^4 + t^3 + 7*t + 49

comment: LMFDB 2.7.b_a. The class is simple, has p-rank 1, and is not ordinary.

2.7.b_b:

t^4 + t^3 + t^2 + 7*t + 49

comment: LMFDB 2.7.b_b. The class is simple, has p-rank 2, and is ordinary.

2.7.b_ab:

t^4 + t^3 - t^2 + 7*t + 49

comment: LMFDB 2.7.b_ab. The class is simple, has p-rank 2, and is ordinary.

2.7.b_c:

t^4 + t^3 + 2*t^2 + 7*t + 49

comment: LMFDB 2.7.b_c. The class is not simple, has p-rank 2, and is ordinary.

2.7.b_ac:

t^4 + t^3 - 2*t^2 + 7*t + 49

comment: LMFDB 2.7.b_ac. The class is simple, has p-rank 2, and is ordinary.

2.7.b_d:

t^4 + t^3 + 3*t^2 + 7*t + 49

comment: LMFDB 2.7.b_d. The class is simple, has p-rank 2, and is ordinary.

2.7.b_ad:

t^4 + t^3 - 3*t^2 + 7*t + 49

comment: LMFDB 2.7.b_ad. The class is simple, has p-rank 2, and is ordinary.

2.7.b_e:

t^4 + t^3 + 4*t^2 + 7*t + 49

comment: LMFDB 2.7.b_e. The class is simple, has p-rank 2, and is ordinary.

2.7.b_ae:

t^4 + t^3 - 4*t^2 + 7*t + 49

comment: LMFDB 2.7.b_ae. The class is simple, has p-rank 2, and is ordinary.

2.7.b_f:

t^4 + t^3 + 5*t^2 + 7*t + 49

comment: LMFDB 2.7.b_f. The class is simple, has p-rank 2, and is ordinary.

2.7.b_af:

t^4 + t^3 - 5*t^2 + 7*t + 49

comment: LMFDB 2.7.b_af. The class is simple, has p-rank 2, and is ordinary.

2.7.b_g:

t^4 + t^3 + 6*t^2 + 7*t + 49

comment: LMFDB 2.7.b_g. The class is simple, has p-rank 2, and is ordinary.

2.7.b_ag:

t^4 + t^3 - 6*t^2 + 7*t + 49

comment: LMFDB 2.7.b_ag. The class is not simple, has p-rank 2, and is ordinary.

2.7.b_h:

t^4 + t^3 + 7*t^2 + 7*t + 49

comment: LMFDB 2.7.b_h. The class is simple, has p-rank 1, and is not ordinary.

2.7.b_ah:

t^4 + t^3 - 7*t^2 + 7*t + 49

comment: LMFDB 2.7.b_ah. The class is simple, has p-rank 1, and is not ordinary.

2.7.b_i:

t^4 + t^3 + 8*t^2 + 7*t + 49

comment: LMFDB 2.7.b_i. The class is not simple, has p-rank 2, and is ordinary.

2.7.b_ai:

t^4 + t^3 - 8*t^2 + 7*t + 49

comment: LMFDB 2.7.b_ai. The class is simple, has p-rank 2, and is ordinary.

2.7.b_j:

t^4 + t^3 + 9*t^2 + 7*t + 49

comment: LMFDB 2.7.b_j. The class is simple, has p-rank 2, and is ordinary.

2.7.b_k:

t^4 + t^3 + 10*t^2 + 7*t + 49

comment: LMFDB 2.7.b_k. The class is simple, has p-rank 2, and is ordinary.

2.7.b_l:

t^4 + t^3 + 11*t^2 + 7*t + 49

comment: LMFDB 2.7.b_l. The class is simple, has p-rank 2, and is ordinary.

2.7.b_m:

t^4 + t^3 + 12*t^2 + 7*t + 49

comment: LMFDB 2.7.b_m. The class is not simple, has p-rank 2, and is ordinary.

2.7.b_n:

t^4 + t^3 + 13*t^2 + 7*t + 49

comment: LMFDB 2.7.b_n. The class is simple, has p-rank 2, and is ordinary.

2.7.b_o:

t^4 + t^3 + 14*t^2 + 7*t + 49

comment: LMFDB 2.7.b_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.ab_a:

t^4 - t^3 - 7*t + 49

comment: LMFDB 2.7.ab_a. The class is simple, has p-rank 1, and is not ordinary.

2.7.ab_b:

t^4 - t^3 + t^2 - 7*t + 49

comment: LMFDB 2.7.ab_b. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_ab:

t^4 - t^3 - t^2 - 7*t + 49

comment: LMFDB 2.7.ab_ab. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_c:

t^4 - t^3 + 2*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_c. The class is not simple, has p-rank 2, and is ordinary.

2.7.ab_ac:

t^4 - t^3 - 2*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_ac. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_d:

t^4 - t^3 + 3*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_d. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_ad:

t^4 - t^3 - 3*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_ad. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_e:

t^4 - t^3 + 4*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_e. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_ae:

t^4 - t^3 - 4*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_ae. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_f:

t^4 - t^3 + 5*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_f. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_af:

t^4 - t^3 - 5*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_af. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_g:

t^4 - t^3 + 6*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_g. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_ag:

t^4 - t^3 - 6*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_ag. The class is not simple, has p-rank 2, and is ordinary.

2.7.ab_h:

t^4 - t^3 + 7*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_h. The class is simple, has p-rank 1, and is not ordinary.

2.7.ab_ah:

t^4 - t^3 - 7*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_ah. The class is simple, has p-rank 1, and is not ordinary.

2.7.ab_i:

t^4 - t^3 + 8*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_i. The class is not simple, has p-rank 2, and is ordinary.

2.7.ab_ai:

t^4 - t^3 - 8*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_ai. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_j:

t^4 - t^3 + 9*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_j. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_k:

t^4 - t^3 + 10*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_k. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_l:

t^4 - t^3 + 11*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_l. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_m:

t^4 - t^3 + 12*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_m. The class is not simple, has p-rank 2, and is ordinary.

2.7.ab_n:

t^4 - t^3 + 13*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_n. The class is simple, has p-rank 2, and is ordinary.

2.7.ab_o:

t^4 - t^3 + 14*t^2 - 7*t + 49

comment: LMFDB 2.7.ab_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.c_a:

t^4 + 2*t^3 + 14*t + 49

comment: LMFDB 2.7.c_a. The class is simple, has p-rank 1, and is not ordinary.

2.7.c_b:

t^4 + 2*t^3 + t^2 + 14*t + 49

comment: LMFDB 2.7.c_b. The class is simple, has p-rank 2, and is ordinary.

2.7.c_ab:

t^4 + 2*t^3 - t^2 + 14*t + 49

comment: LMFDB 2.7.c_ab. The class is not simple, has p-rank 2, and is ordinary.

2.7.c_c:

t^4 + 2*t^3 + 2*t^2 + 14*t + 49

comment: LMFDB 2.7.c_c. The class is simple, has p-rank 2, and is ordinary.

2.7.c_ac:

t^4 + 2*t^3 - 2*t^2 + 14*t + 49

comment: LMFDB 2.7.c_ac. The class is simple, has p-rank 2, and is ordinary.

2.7.c_d:

t^4 + 2*t^3 + 3*t^2 + 14*t + 49

comment: LMFDB 2.7.c_d. The class is simple, has p-rank 2, and is ordinary.

2.7.c_ad:

t^4 + 2*t^3 - 3*t^2 + 14*t + 49

comment: LMFDB 2.7.c_ad. The class is simple, has p-rank 2, and is ordinary.

2.7.c_e:

t^4 + 2*t^3 + 4*t^2 + 14*t + 49

comment: LMFDB 2.7.c_e. The class is simple, has p-rank 2, and is ordinary.

2.7.c_f:

t^4 + 2*t^3 + 5*t^2 + 14*t + 49

comment: LMFDB 2.7.c_f. The class is simple, has p-rank 2, and is ordinary.

2.7.c_g:

t^4 + 2*t^3 + 6*t^2 + 14*t + 49

comment: LMFDB 2.7.c_g. The class is not simple, has p-rank 2, and is ordinary.

2.7.c_h:

t^4 + 2*t^3 + 7*t^2 + 14*t + 49

comment: LMFDB 2.7.c_h. The class is simple, has p-rank 1, and is not ordinary.

2.7.c_i:

t^4 + 2*t^3 + 8*t^2 + 14*t + 49

comment: LMFDB 2.7.c_i. The class is simple, has p-rank 2, and is ordinary.

2.7.c_j:

t^4 + 2*t^3 + 9*t^2 + 14*t + 49

comment: LMFDB 2.7.c_j. The class is simple, has p-rank 2, and is ordinary.

2.7.c_k:

t^4 + 2*t^3 + 10*t^2 + 14*t + 49

comment: LMFDB 2.7.c_k. The class is simple, has p-rank 2, and is ordinary.

2.7.c_l:

t^4 + 2*t^3 + 11*t^2 + 14*t + 49

comment: LMFDB 2.7.c_l. The class is not simple, has p-rank 2, and is ordinary.

2.7.c_m:

t^4 + 2*t^3 + 12*t^2 + 14*t + 49

comment: LMFDB 2.7.c_m. The class is simple, has p-rank 2, and is ordinary.

2.7.c_n:

t^4 + 2*t^3 + 13*t^2 + 14*t + 49

comment: LMFDB 2.7.c_n. The class is simple, has p-rank 2, and is ordinary.

2.7.c_o:

t^4 + 2*t^3 + 14*t^2 + 14*t + 49

comment: LMFDB 2.7.c_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.c_p:

t^4 + 2*t^3 + 15*t^2 + 14*t + 49

comment: LMFDB 2.7.c_p. The class is not simple, has p-rank 2, and is ordinary.

2.7.ac_a:

t^4 - 2*t^3 - 14*t + 49

comment: LMFDB 2.7.ac_a. The class is simple, has p-rank 1, and is not ordinary.

2.7.ac_b:

t^4 - 2*t^3 + t^2 - 14*t + 49

comment: LMFDB 2.7.ac_b. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_ab:

t^4 - 2*t^3 - t^2 - 14*t + 49

comment: LMFDB 2.7.ac_ab. The class is not simple, has p-rank 2, and is ordinary.

2.7.ac_c:

t^4 - 2*t^3 + 2*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_c. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_ac:

t^4 - 2*t^3 - 2*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_ac. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_d:

t^4 - 2*t^3 + 3*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_d. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_ad:

t^4 - 2*t^3 - 3*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_ad. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_e:

t^4 - 2*t^3 + 4*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_e. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_f:

t^4 - 2*t^3 + 5*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_f. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_g:

t^4 - 2*t^3 + 6*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_g. The class is not simple, has p-rank 2, and is ordinary.

2.7.ac_h:

t^4 - 2*t^3 + 7*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_h. The class is simple, has p-rank 1, and is not ordinary.

2.7.ac_i:

t^4 - 2*t^3 + 8*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_i. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_j:

t^4 - 2*t^3 + 9*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_j. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_k:

t^4 - 2*t^3 + 10*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_k. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_l:

t^4 - 2*t^3 + 11*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_l. The class is not simple, has p-rank 2, and is ordinary.

2.7.ac_m:

t^4 - 2*t^3 + 12*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_m. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_n:

t^4 - 2*t^3 + 13*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_n. The class is simple, has p-rank 2, and is ordinary.

2.7.ac_o:

t^4 - 2*t^3 + 14*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.ac_p:

t^4 - 2*t^3 + 15*t^2 - 14*t + 49

comment: LMFDB 2.7.ac_p. The class is not simple, has p-rank 2, and is ordinary.

2.7.d_c:

t^4 + 3*t^3 + 2*t^2 + 21*t + 49

comment: LMFDB 2.7.d_c. The class is simple, has p-rank 2, and is ordinary.

2.7.d_d:

t^4 + 3*t^3 + 3*t^2 + 21*t + 49

comment: LMFDB 2.7.d_d. The class is simple, has p-rank 2, and is ordinary.

2.7.d_e:

t^4 + 3*t^3 + 4*t^2 + 21*t + 49

comment: LMFDB 2.7.d_e. The class is not simple, has p-rank 2, and is ordinary.

2.7.d_f:

t^4 + 3*t^3 + 5*t^2 + 21*t + 49

comment: LMFDB 2.7.d_f. The class is simple, has p-rank 2, and is ordinary.

2.7.d_g:

t^4 + 3*t^3 + 6*t^2 + 21*t + 49

comment: LMFDB 2.7.d_g. The class is simple, has p-rank 2, and is ordinary.

2.7.d_h:

t^4 + 3*t^3 + 7*t^2 + 21*t + 49

comment: LMFDB 2.7.d_h. The class is simple, has p-rank 1, and is not ordinary.

2.7.d_i:

t^4 + 3*t^3 + 8*t^2 + 21*t + 49

comment: LMFDB 2.7.d_i. The class is simple, has p-rank 2, and is ordinary.

2.7.d_j:

t^4 + 3*t^3 + 9*t^2 + 21*t + 49

comment: LMFDB 2.7.d_j. The class is simple, has p-rank 2, and is ordinary.

2.7.d_k:

t^4 + 3*t^3 + 10*t^2 + 21*t + 49

comment: LMFDB 2.7.d_k. The class is not simple, has p-rank 2, and is ordinary.

2.7.d_l:

t^4 + 3*t^3 + 11*t^2 + 21*t + 49

comment: LMFDB 2.7.d_l. The class is simple, has p-rank 2, and is ordinary.

2.7.d_m:

t^4 + 3*t^3 + 12*t^2 + 21*t + 49

comment: LMFDB 2.7.d_m. The class is simple, has p-rank 2, and is ordinary.

2.7.d_n:

t^4 + 3*t^3 + 13*t^2 + 21*t + 49

comment: LMFDB 2.7.d_n. The class is simple, has p-rank 2, and is ordinary.

2.7.d_o:

t^4 + 3*t^3 + 14*t^2 + 21*t + 49

comment: LMFDB 2.7.d_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.d_p:

t^4 + 3*t^3 + 15*t^2 + 21*t + 49

comment: LMFDB 2.7.d_p. The class is simple, has p-rank 2, and is ordinary.

2.7.d_q:

t^4 + 3*t^3 + 16*t^2 + 21*t + 49

comment: LMFDB 2.7.d_q. The class is not simple, has p-rank 2, and is ordinary.

2.7.ad_c:

t^4 - 3*t^3 + 2*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_c. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_d:

t^4 - 3*t^3 + 3*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_d. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_e:

t^4 - 3*t^3 + 4*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_e. The class is not simple, has p-rank 2, and is ordinary.

2.7.ad_f:

t^4 - 3*t^3 + 5*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_f. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_g:

t^4 - 3*t^3 + 6*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_g. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_h:

t^4 - 3*t^3 + 7*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_h. The class is simple, has p-rank 1, and is not ordinary.

2.7.ad_i:

t^4 - 3*t^3 + 8*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_i. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_j:

t^4 - 3*t^3 + 9*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_j. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_k:

t^4 - 3*t^3 + 10*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_k. The class is not simple, has p-rank 2, and is ordinary.

2.7.ad_l:

t^4 - 3*t^3 + 11*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_l. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_m:

t^4 - 3*t^3 + 12*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_m. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_n:

t^4 - 3*t^3 + 13*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_n. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_o:

t^4 - 3*t^3 + 14*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.ad_p:

t^4 - 3*t^3 + 15*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_p. The class is simple, has p-rank 2, and is ordinary.

2.7.ad_q:

t^4 - 3*t^3 + 16*t^2 - 21*t + 49

comment: LMFDB 2.7.ad_q. The class is not simple, has p-rank 2, and is ordinary.

2.7.e_i:

t^4 + 4*t^3 + 8*t^2 + 28*t + 49

comment: LMFDB 2.7.e_i. The class is simple, has p-rank 2, and is ordinary.

2.7.e_j:

t^4 + 4*t^3 + 9*t^2 + 28*t + 49

comment: LMFDB 2.7.e_j. The class is not simple, has p-rank 2, and is ordinary.

2.7.e_k:

t^4 + 4*t^3 + 10*t^2 + 28*t + 49

comment: LMFDB 2.7.e_k. The class is simple, has p-rank 2, and is ordinary.

2.7.e_l:

t^4 + 4*t^3 + 11*t^2 + 28*t + 49

comment: LMFDB 2.7.e_l. The class is simple, has p-rank 2, and is ordinary.

2.7.e_m:

t^4 + 4*t^3 + 12*t^2 + 28*t + 49

comment: LMFDB 2.7.e_m. The class is simple, has p-rank 2, and is ordinary.

2.7.e_n:

t^4 + 4*t^3 + 13*t^2 + 28*t + 49

comment: LMFDB 2.7.e_n. The class is simple, has p-rank 2, and is ordinary.

2.7.e_o:

t^4 + 4*t^3 + 14*t^2 + 28*t + 49

comment: LMFDB 2.7.e_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.e_p:

t^4 + 4*t^3 + 15*t^2 + 28*t + 49

comment: LMFDB 2.7.e_p. The class is simple, has p-rank 2, and is ordinary.

2.7.e_q:

t^4 + 4*t^3 + 16*t^2 + 28*t + 49

comment: LMFDB 2.7.e_q. The class is simple, has p-rank 2, and is ordinary.

2.7.e_r:

t^4 + 4*t^3 + 17*t^2 + 28*t + 49

comment: LMFDB 2.7.e_r. The class is not simple, has p-rank 2, and is ordinary.

2.7.e_s:

t^4 + 4*t^3 + 18*t^2 + 28*t + 49

comment: LMFDB 2.7.e_s. The class is not simple, has p-rank 2, and is ordinary.

2.7.ae_i:

t^4 - 4*t^3 + 8*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_i. The class is simple, has p-rank 2, and is ordinary.

2.7.ae_j:

t^4 - 4*t^3 + 9*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_j. The class is not simple, has p-rank 2, and is ordinary.

2.7.ae_k:

t^4 - 4*t^3 + 10*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_k. The class is simple, has p-rank 2, and is ordinary.

2.7.ae_l:

t^4 - 4*t^3 + 11*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_l. The class is simple, has p-rank 2, and is ordinary.

2.7.ae_m:

t^4 - 4*t^3 + 12*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_m. The class is simple, has p-rank 2, and is ordinary.

2.7.ae_n:

t^4 - 4*t^3 + 13*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_n. The class is simple, has p-rank 2, and is ordinary.

2.7.ae_o:

t^4 - 4*t^3 + 14*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.ae_p:

t^4 - 4*t^3 + 15*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_p. The class is simple, has p-rank 2, and is ordinary.

2.7.ae_q:

t^4 - 4*t^3 + 16*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_q. The class is simple, has p-rank 2, and is ordinary.

2.7.ae_r:

t^4 - 4*t^3 + 17*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_r. The class is not simple, has p-rank 2, and is ordinary.

2.7.ae_s:

t^4 - 4*t^3 + 18*t^2 - 28*t + 49

comment: LMFDB 2.7.ae_s. The class is not simple, has p-rank 2, and is ordinary.

2.7.f_n:

t^4 + 5*t^3 + 13*t^2 + 35*t + 49

comment: LMFDB 2.7.f_n. The class is simple, has p-rank 2, and is ordinary.

2.7.f_o:

t^4 + 5*t^3 + 14*t^2 + 35*t + 49

comment: LMFDB 2.7.f_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.f_p:

t^4 + 5*t^3 + 15*t^2 + 35*t + 49

comment: LMFDB 2.7.f_p. The class is simple, has p-rank 2, and is ordinary.

2.7.f_q:

t^4 + 5*t^3 + 16*t^2 + 35*t + 49

comment: LMFDB 2.7.f_q. The class is simple, has p-rank 2, and is ordinary.

2.7.f_r:

t^4 + 5*t^3 + 17*t^2 + 35*t + 49

comment: LMFDB 2.7.f_r. The class is simple, has p-rank 2, and is ordinary.

2.7.f_s:

t^4 + 5*t^3 + 18*t^2 + 35*t + 49

comment: LMFDB 2.7.f_s. The class is not simple, has p-rank 2, and is ordinary.

2.7.f_t:

t^4 + 5*t^3 + 19*t^2 + 35*t + 49

comment: LMFDB 2.7.f_t. The class is simple, has p-rank 2, and is ordinary.

2.7.f_u:

t^4 + 5*t^3 + 20*t^2 + 35*t + 49

comment: LMFDB 2.7.f_u. The class is not simple, has p-rank 2, and is ordinary.

2.7.af_n:

t^4 - 5*t^3 + 13*t^2 - 35*t + 49

comment: LMFDB 2.7.af_n. The class is simple, has p-rank 2, and is ordinary.

2.7.af_o:

t^4 - 5*t^3 + 14*t^2 - 35*t + 49

comment: LMFDB 2.7.af_o. The class is not simple, has p-rank 1, and is not ordinary.

2.7.af_p:

t^4 - 5*t^3 + 15*t^2 - 35*t + 49

comment: LMFDB 2.7.af_p. The class is simple, has p-rank 2, and is ordinary.

2.7.af_q:

t^4 - 5*t^3 + 16*t^2 - 35*t + 49

comment: LMFDB 2.7.af_q. The class is simple, has p-rank 2, and is ordinary.

2.7.af_r:

t^4 - 5*t^3 + 17*t^2 - 35*t + 49

comment: LMFDB 2.7.af_r. The class is simple, has p-rank 2, and is ordinary.

2.7.af_s:

t^4 - 5*t^3 + 18*t^2 - 35*t + 49

comment: LMFDB 2.7.af_s. The class is not simple, has p-rank 2, and is ordinary.

2.7.af_t:

t^4 - 5*t^3 + 19*t^2 - 35*t + 49

comment: LMFDB 2.7.af_t. The class is simple, has p-rank 2, and is ordinary.

2.7.af_u:

t^4 - 5*t^3 + 20*t^2 - 35*t + 49

comment: LMFDB 2.7.af_u. The class is not simple, has p-rank 2, and is ordinary.

2.7.g_s:

t^4 + 6*t^3 + 18*t^2 + 42*t + 49

comment: LMFDB 2.7.g_s. The class is simple, has p-rank 2, and is ordinary.

2.7.g_t:

t^4 + 6*t^3 + 19*t^2 + 42*t + 49

comment: LMFDB 2.7.g_t. The class is not simple, has p-rank 2, and is ordinary.

2.7.g_u:

t^4 + 6*t^3 + 20*t^2 + 42*t + 49

comment: LMFDB 2.7.g_u. The class is simple, has p-rank 2, and is ordinary.

2.7.g_v:

t^4 + 6*t^3 + 21*t^2 + 42*t + 49

comment: LMFDB 2.7.g_v. The class is simple, has p-rank 1, and is not ordinary.

2.7.g_w:

t^4 + 6*t^3 + 22*t^2 + 42*t + 49

comment: LMFDB 2.7.g_w. The class is not simple, has p-rank 2, and is ordinary.

2.7.g_x:

t^4 + 6*t^3 + 23*t^2 + 42*t + 49

comment: LMFDB 2.7.g_x. The class is not simple, has p-rank 2, and is ordinary.

2.7.ag_s:

t^4 - 6*t^3 + 18*t^2 - 42*t + 49

comment: LMFDB 2.7.ag_s. The class is simple, has p-rank 2, and is ordinary.

2.7.ag_t:

t^4 - 6*t^3 + 19*t^2 - 42*t + 49

comment: LMFDB 2.7.ag_t. The class is not simple, has p-rank 2, and is ordinary.

2.7.ag_u:

t^4 - 6*t^3 + 20*t^2 - 42*t + 49

comment: LMFDB 2.7.ag_u. The class is simple, has p-rank 2, and is ordinary.

2.7.ag_v:

t^4 - 6*t^3 + 21*t^2 - 42*t + 49

comment: LMFDB 2.7.ag_v. The class is simple, has p-rank 1, and is not ordinary.

2.7.ag_w:

t^4 - 6*t^3 + 22*t^2 - 42*t + 49

comment: LMFDB 2.7.ag_w. The class is not simple, has p-rank 2, and is ordinary.

2.7.ag_x:

t^4 - 6*t^3 + 23*t^2 - 42*t + 49

comment: LMFDB 2.7.ag_x. The class is not simple, has p-rank 2, and is ordinary.

2.7.h_y:

t^4 + 7*t^3 + 24*t^2 + 49*t + 49

comment: LMFDB 2.7.h_y. The class is not simple, has p-rank 2, and is ordinary.

2.7.h_z:

t^4 + 7*t^3 + 25*t^2 + 49*t + 49

comment: LMFDB 2.7.h_z. The class is simple, has p-rank 2, and is ordinary.

2.7.h_ba:

t^4 + 7*t^3 + 26*t^2 + 49*t + 49

comment: LMFDB 2.7.h_ba. The class is not simple, has p-rank 2, and is ordinary.

2.7.ah_y:

t^4 - 7*t^3 + 24*t^2 - 49*t + 49

comment: LMFDB 2.7.ah_y. The class is not simple, has p-rank 2, and is ordinary.

2.7.ah_z:

t^4 - 7*t^3 + 25*t^2 - 49*t + 49

comment: LMFDB 2.7.ah_z. The class is simple, has p-rank 2, and is ordinary.

2.7.ah_ba:

t^4 - 7*t^3 + 26*t^2 - 49*t + 49

comment: LMFDB 2.7.ah_ba. The class is not simple, has p-rank 2, and is ordinary.

2.7.i_bd:

t^4 + 8*t^3 + 29*t^2 + 56*t + 49

comment: LMFDB 2.7.i_bd. The class is not simple, has p-rank 2, and is ordinary.

2.7.i_be:

t^4 + 8*t^3 + 30*t^2 + 56*t + 49

comment: LMFDB 2.7.i_be. The class is not simple, has p-rank 2, and is ordinary.

2.7.ai_bd:

t^4 - 8*t^3 + 29*t^2 - 56*t + 49

comment: LMFDB 2.7.ai_bd. The class is not simple, has p-rank 2, and is ordinary.

2.7.ai_be:

t^4 - 8*t^3 + 30*t^2 - 56*t + 49

comment: LMFDB 2.7.ai_be. The class is not simple, has p-rank 2, and is ordinary.

2.7.j_bi:

t^4 + 9*t^3 + 34*t^2 + 63*t + 49

comment: LMFDB 2.7.j_bi. The class is not simple, has p-rank 2, and is ordinary.

2.7.aj_bi:

t^4 - 9*t^3 + 34*t^2 - 63*t + 49

comment: LMFDB 2.7.aj_bi. The class is not simple, has p-rank 2, and is ordinary.

2.7.k_bn:

t^4 + 10*t^3 + 39*t^2 + 70*t + 49

comment: LMFDB 2.7.k_bn. The class is not simple, has p-rank 2, and is ordinary.

2.7.ak_bn:

t^4 - 10*t^3 + 39*t^2 - 70*t + 49

comment: LMFDB 2.7.ak_bn. The class is not simple, has p-rank 2, and is ordinary.

Definition

The Weil polynomial of an abelian variety $A/\mathbb{F}_q$ is $P_A(t)=\det(t-F\mid T_\ell A)$, where $F$ is the $q$-power Frobenius and $\ell\nmid q$ [3]. This table gives $P_A(t)$ for isogeny classes over prime fields.

Parameters

$g$

— dimension ($g\geq1$)

$q$

— base-field cardinality ($q$ is prime)

label

— LMFDB isogeny-class label

Formulas

(1)

$L_A(x)=x^{2g}P_A(1/x)$ and $P_A(t)=t^{2g}L_A(1/t)$, so the roots of $P_A$ have absolute value $\sqrt q$ while the roots of $L_A$ have absolute value $q^{-1/2}$ [4].

(2)

$P_A(t)=t^{2g}+a_1t^{2g-1}+\cdots+a_gt^g+ qa_{g-1}t^{g-1}+\cdots+q^{g-1}a_1t+q^g$.

(3)

The final part of the LMFDB label encodes $a_1,\ldots,a_g$ from (2) in base $26$, with $a=0$, $b=1$, and an initial $a$ marking a negative integer [5].

(4)

For $q=p$ prime, the $p$-rank is the largest $r$ with $0\leq r\leq g$ such that $p\nmid a_r$, where $a_0=1$ [6].

(5)

For $g=1$, $P_A(t)=t^2-a_qt+q$, where $a_q=q+1-\#E(\mathbb{F}_q)$ for an elliptic curve $E$ in the isogeny class.

(6)

For $g=2$, $P_A(t)=t^4+a_1t^3+a_2t^2+qa_1t+q^2$ is a Weil $q$-polynomial exactly when $|a_1|\leq4\sqrt q$ and $2|a_1|\sqrt q-2q\leq a_2\leq a_1^2/4+2q$.

Comments

(7)

The polynomial stored here is the characteristic polynomial $P_A(t)$. The L-polynomial $L_A(x)$ used on LMFDB class pages is its reverse, as in (1).

(8)

The label is the LMFDB label for the isogeny class [5]. It identifies the isogeny class, which $g$ and $q$ alone do not.

(9)

The table is restricted to prime fields. For $g=1$ and prime $q$, every integer $a_q$ with $a_q^2\leq4q$ occurs as the trace of an elliptic curve [1]; for $g=2$, the enumeration by (6) matches the LMFDB count of surface isogeny classes for each $q$ in the table [7] [8] [9]. Over proper prime powers, extra occurrence conditions enter; for example, the Hasse-admissible traces $\pm2$ do not occur over $\mathbb{F}_8$.

(10)

Each entry comment says whether the isogeny class is simple, its $p$-rank, and whether it is ordinary [2]. The question whether a genus $2$ class contains a Jacobian is not recorded here.

Programs

(P1)

Sage

R.<t> = ZZ[]
q = 17
[t^2 + a*t + q for a in range(-floor(2*sqrt(q)), floor(2*sqrt(q)) + 1)]

(P2)

Sage

R.<t> = ZZ[]
q = 11
polys = []
for a1 in range(-floor(4*sqrt(q)), floor(4*sqrt(q)) + 1):
    lo = ceil(2*abs(a1)*sqrt(q) - 2*q)
    hi = floor(a1^2/4 + 2*q)
    for a2 in range(lo, hi + 1):
        polys.append(t^4 + a1*t^3 + a2*t^2 + q*a1*t + q^2)
polys

References

[1]

William C. Waterhouse, Abelian varieties over finite fields, Annales scientifiques de l'Ecole Normale Superieure, Serie 4, 2 (1969), 521-560. (doi) (MR)

[2]

Taylor Dupuy, Kiran S. Kedlaya, David Roe and Christelle Vincent, Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB, Arithmetic Geometry, Number Theory, and Computation, Simons Symposia, Springer, 2021, 375-448. (arXiv)

Links

[3]

LMFDB knowl: characteristic polynomial of an abelian variety over a finite field

[4]

LMFDB knowl: L-polynomial of an abelian variety over a finite field

[5]

LMFDB knowl: labels for isogeny classes of abelian varieties over finite fields

[6]

LMFDB knowl: p-rank

[7]

LMFDB: isogeny classes of abelian varieties over finite fields

[8]

LMFDB: source and acknowledgments for Weil polynomial data

[9]

LMFDB: completeness of Weil polynomial data

[10]

OEIS A362198: number of isogeny classes of abelian surfaces over finite fields

Similar tables

Gauss sums of primitive Dirichlet characters — stores finite-field character sums whose values are Frobenius eigenvalue sums in related point-counting problems

Jacobi sums of pairs of Dirichlet characters modulo a prime — stores finite-field character sums that can determine Frobenius polynomials of diagonal varieties

Kloosterman sums modulo a prime — stores another family of Frobenius trace sums over finite fields

Roots of unity — contains the normalised roots $\alpha/\sqrt q$ when a Weil polynomial is supersingular

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every isogeny class for $g=1$ and prime $q\leq13$, and every isogeny class for $g=2$ and prime $q\leq7$)

How they were obtained:

The generator enumerates the coefficient tuples in (2) exactly. For $g=1$ it uses the Hasse bound $a_1^2\leq4q$, giving $5,7,9,11,13,15$ classes for $q=2,3,5,7,11,13$.

For $g=2$ it uses (6), giving $35,63,129,207$ classes for $q=2,3,5,7$, which matches OEIS A362198 [10] and the LMFDB source and completeness pages [8] [9]. Before an entry is returned, the generator requires the polynomial to be monic, to have degree $2g$, to satisfy the $q$-reciprocity in (2), to satisfy the coefficient bounds for its dimension, and to have the LMFDB label and the $p$-rank stated in the entry comment. The simplicity statement in each entry comment is determined from the stored polynomial by decomposition into elliptic Weil polynomials; the four classes with polynomial $(t^2-q)^2$ are the simple supersingular Honda-Tate classes.