Frobenius polynomials of hypergeometric Calabi–Yau motives

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Polynomials

$A$

$z$

$p$

$P_{A,z,p}(T)$

2.2.2.2

1/2

729*T^4 + 6*T^2 + 1

2.2.2.2

1/2

15625*T^4 + 1500*T^3 + 190*T^2 + 12*T + 1

2.2.2.2

1/2

117649*T^4 + 5488*T^3 + 238*T^2 + 16*T + 1

2.2.2.2

1/2

11:

1771561*T^4 + 85184*T^3 + 2486*T^2 + 64*T + 1

2.2.2.2

1/2

13:

4826809*T^4 - 8788*T^3 - 1586*T^2 - 4*T + 1

2.2.2.2

1/2

17:

24137569*T^4 - 176868*T^3 + 1190*T^2 - 36*T + 1

2.2.2.2

1/2

19:

47045881*T^4 + 1254*T^2 + 1

2.2.2.2

1/2

23:

148035889*T^4 + 2141392*T^3 + 22862*T^2 + 176*T + 1

2.2.2.2

1/2

29:

594823321*T^4 - 3999796*T^3 + 28014*T^2 - 164*T + 1

2.2.2.2

1/2

31:

887503681*T^4 - 1906624*T^3 - 3906*T^2 - 64*T + 1

2.2.2.2

-1

729*T^4 + 6*T^2 + 1

2.2.2.2

-1

15625*T^4 + 500*T^3 - 130*T^2 + 4*T + 1

2.2.2.2

-1

117649*T^4 + 238*T^2 + 1

2.2.2.2

-1

11:

1771561*T^4 + 2486*T^2 + 1

2.2.2.2

-1

13:

4826809*T^4 + 184548*T^3 + 4238*T^2 + 84*T + 1

2.2.2.2

-1

17:

24137569*T^4 - 176868*T^3 - 3162*T^2 - 36*T + 1

2.2.2.2

-1

19:

47045881*T^4 + 10982*T^2 + 1

2.2.2.2

-1

23:

148035889*T^4 - 12466*T^2 + 1

2.2.2.2

-1

29:

594823321*T^4 - 3414460*T^3 + 41006*T^2 - 140*T + 1

2.2.2.2

-1

31:

887503681*T^4 + 27838*T^2 + 1

2.2.2.2

729*T^4 + 6*T^2 + 1

2.2.2.2

15625*T^4 - 1500*T^3 + 190*T^2 - 12*T + 1

2.2.2.2

117649*T^4 + 5488*T^3 + 238*T^2 + 16*T + 1

2.2.2.2

11:

1771561*T^4 - 85184*T^3 + 2486*T^2 - 64*T + 1

2.2.2.2

13:

4826809*T^4 + 8788*T^3 - 1586*T^2 + 4*T + 1

2.2.2.2

17:

24137569*T^4 - 176868*T^3 + 1190*T^2 - 36*T + 1

2.2.2.2

19:

47045881*T^4 + 1254*T^2 + 1

2.2.2.2

23:

148035889*T^4 + 2141392*T^3 + 22862*T^2 + 176*T + 1

2.2.2.2

29:

594823321*T^4 + 3999796*T^3 + 28014*T^2 + 164*T + 1

2.2.2.2

31:

887503681*T^4 - 1906624*T^3 - 3906*T^2 - 64*T + 1

2.2.3

1/2

15625*T^4 - 250*T^3 - 110*T^2 - 2*T + 1

2.2.3

1/2

117649*T^4 + 5488*T^3 + 350*T^2 + 16*T + 1

2.2.3

1/2

11:

1771561*T^4 + 15972*T^3 - 506*T^2 + 12*T + 1

2.2.3

1/2

13:

4826809*T^4 + 8788*T^3 - 130*T^2 + 4*T + 1

2.2.3

1/2

17:

24137569*T^4 - 658342*T^3 + 12274*T^2 - 134*T + 1

2.2.3

1/2

19:

47045881*T^4 + 850516*T^3 + 10070*T^2 + 124*T + 1

2.2.3

1/2

23:

148035889*T^4 - 1460040*T^3 + 11086*T^2 - 120*T + 1

2.2.3

1/2

29:

594823321*T^4 + 1512118*T^3 + 34162*T^2 + 62*T + 1

2.2.3

1/2

31:

887503681*T^4 - 1668296*T^3 + 10478*T^2 - 56*T + 1

2.2.3

-1

15625*T^4 + 1375*T^3 + 160*T^2 + 11*T + 1

2.2.3

-1

117649*T^4 - 3773*T^3 - 154*T^2 - 11*T + 1

2.2.3

-1

11:

1771561*T^4 - 27951*T^3 + 682*T^2 - 21*T + 1

2.2.3

-1

13:

4826809*T^4 + 74698*T^3 + 3770*T^2 + 34*T + 1

2.2.3

-1

17:

24137569*T^4 + 579734*T^3 + 8602*T^2 + 118*T + 1

2.2.3

-1

19:

47045881*T^4 - 836798*T^3 + 13262*T^2 - 122*T + 1

2.2.3

-1

23:

148035889*T^4 - 1241034*T^3 + 11086*T^2 - 102*T + 1

2.2.3

-1

29:

594823321*T^4 + 8584928*T^3 + 68614*T^2 + 352*T + 1

2.2.3

-1

31:

887503681*T^4 + 2353489*T^3 - 2914*T^2 + 79*T + 1

2.2.3

15625*T^4 + 625*T^3 + 160*T^2 + 5*T + 1

2.2.3

117649*T^4 - 3773*T^3 + 182*T^2 - 11*T + 1

2.2.3

11:

1771561*T^4 + 59895*T^3 + 1474*T^2 + 45*T + 1

2.2.3

13:

4826809*T^4 - 4394*T^3 + 650*T^2 - 2*T + 1

2.2.3

17:

24137569*T^4 + 108086*T^3 + 3706*T^2 + 22*T + 1

2.2.3

19:

47045881*T^4 - 260642*T^3 + 6878*T^2 - 38*T + 1

2.2.3

23:

148035889*T^4 + 1971054*T^3 + 21022*T^2 + 162*T + 1

2.2.3

29:

594823321*T^4 + 2731568*T^3 + 35206*T^2 + 112*T + 1

2.2.3

31:

887503681*T^4 + 2353489*T^3 - 2914*T^2 + 79*T + 1

2.2.4

1/2

729*T^4 + 54*T^3 - 18*T^2 + 2*T + 1

2.2.4

1/2

15625*T^4 - 250*T^3 + 90*T^2 - 2*T + 1

2.2.4

1/2

117649*T^4 + 2744*T^3 - 210*T^2 + 8*T + 1

2.2.4

1/2

11:

1771561*T^4 - 13310*T^3 - 418*T^2 - 10*T + 1

2.2.4

1/2

13:

4826809*T^4 + 118638*T^3 + 2314*T^2 + 54*T + 1

2.2.4

1/2

17:

24137569*T^4 - 530604*T^3 + 12342*T^2 - 108*T + 1

2.2.4

1/2

19:

47045881*T^4 + 342950*T^3 + 11438*T^2 + 50*T + 1

2.2.4

1/2

23:

148035889*T^4 - 1265368*T^3 + 18446*T^2 - 104*T + 1

2.2.4

1/2

29:

594823321*T^4 - 4536354*T^3 + 10730*T^2 - 186*T + 1

2.2.4

1/2

31:

887503681*T^4 - 953312*T^3 + 4030*T^2 - 32*T + 1

2.2.4

-1

729*T^4 + 54*T^3 - 18*T^2 + 2*T + 1

2.2.4

-1

15625*T^4 - 500*T^3 + 110*T^2 - 4*T + 1

2.2.4

-1

117649*T^4 - 4116*T^3 + 462*T^2 - 12*T + 1

2.2.4

-1

11:

1771561*T^4 + 39930*T^3 + 1342*T^2 + 30*T + 1

2.2.4

-1

13:

4826809*T^4 - 26364*T^3 + 1742*T^2 - 12*T + 1

2.2.4

-1

17:

24137569*T^4 + 393040*T^3 + 4862*T^2 + 80*T + 1

2.2.4

-1

19:

47045881*T^4 + 342950*T^3 + 4142*T^2 + 50*T + 1

2.2.4

-1

23:

148035889*T^4 - 243340*T^3 + 2990*T^2 - 20*T + 1

2.2.4

-1

29:

594823321*T^4 + 4585132*T^3 + 52606*T^2 + 188*T + 1

2.2.4

-1

31:

887503681*T^4 - 6673184*T^3 + 19902*T^2 - 224*T + 1

2.2.4

729*T^4 + 54*T^3 - 18*T^2 + 2*T + 1

2.2.4

15625*T^4 - 90*T^2 + 1

2.2.4

117649*T^4 - 9604*T^3 + 462*T^2 - 28*T + 1

2.2.4

11:

1771561*T^4 - 87846*T^3 + 2750*T^2 - 66*T + 1

2.2.4

13:

4826809*T^4 + 158184*T^3 + 4134*T^2 + 72*T + 1

2.2.4

17:

24137569*T^4 + 393040*T^3 + 7038*T^2 + 80*T + 1

2.2.4

19:

47045881*T^4 - 535002*T^3 + 11438*T^2 - 78*T + 1

2.2.4

23:

148035889*T^4 + 730020*T^3 - 5842*T^2 + 60*T + 1

2.2.4

29:

594823321*T^4 + 2731568*T^3 + 23606*T^2 + 112*T + 1

2.2.4

31:

887503681*T^4 + 953312*T^3 - 27714*T^2 + 32*T + 1

2.2.6

1/2

15625*T^4 - 750*T^3 + 10*T^2 - 6*T + 1

2.2.6

1/2

117649*T^4 - 6860*T^3 + 574*T^2 - 20*T + 1

2.2.6

1/2

11:

1771561*T^4 - 26620*T^3 + 902*T^2 - 20*T + 1

2.2.6

1/2

13:

4826809*T^4 - 35152*T^3 + 3718*T^2 - 16*T + 1

2.2.6

1/2

17:

24137569*T^4 + 88434*T^3 + 5474*T^2 + 18*T + 1

2.2.6

1/2

19:

47045881*T^4 - 82308*T^3 + 5814*T^2 - 12*T + 1

2.2.6

1/2

23:

148035889*T^4 + 97336*T^3 + 22862*T^2 + 8*T + 1

2.2.6

1/2

29:

594823321*T^4 - 3853462*T^3 + 38106*T^2 - 158*T + 1

2.2.6

1/2

31:

887503681*T^4 + 2025788*T^3 + 42222*T^2 + 68*T + 1

2.2.6

-1

15625*T^4 - 1375*T^3 + 140*T^2 - 11*T + 1

2.2.6

-1

117649*T^4 + 3087*T^3 + 70*T^2 + 9*T + 1

2.2.6

-1

11:

1771561*T^4 - 67881*T^3 + 1826*T^2 - 51*T + 1

2.2.6

-1

13:

4826809*T^4 + 13182*T^3 + 2210*T^2 + 6*T + 1

2.2.6

-1

17:

24137569*T^4 - 29478*T^3 - 5406*T^2 - 6*T + 1

2.2.6

-1

19:

47045881*T^4 - 1028850*T^3 + 12350*T^2 - 150*T + 1

2.2.6

-1

23:

148035889*T^4 + 3431094*T^3 + 40526*T^2 + 282*T + 1

2.2.6

-1

29:

594823321*T^4 - 3414460*T^3 + 39614*T^2 - 140*T + 1

2.2.6

-1

31:

887503681*T^4 - 5630499*T^3 + 24862*T^2 - 189*T + 1

2.2.6

15625*T^4 + 1125*T^3 + 40*T^2 + 9*T + 1

2.2.6

117649*T^4 + 2401*T^3 + 70*T^2 + 7*T + 1

2.2.6

11:

1771561*T^4 + 14641*T^3 - 1078*T^2 + 11*T + 1

2.2.6

13:

4826809*T^4 + 21970*T^3 - 494*T^2 + 10*T + 1

2.2.6

17:

24137569*T^4 + 442170*T^3 + 7106*T^2 + 90*T + 1

2.2.6

19:

47045881*T^4 + 123462*T^3 - 1026*T^2 + 18*T + 1

2.2.6

23:

148035889*T^4 - 1143698*T^3 + 10718*T^2 - 94*T + 1

2.2.6

29:

594823321*T^4 + 1365784*T^3 + 2262*T^2 + 56*T + 1

2.2.6

31:

887503681*T^4 + 3723875*T^3 - 12834*T^2 + 125*T + 1

3.3

1/2

15625*T^4 - 2250*T^3 + 295*T^2 - 18*T + 1

3.3

1/2

117649*T^4 - 2401*T^3 + 84*T^2 - 7*T + 1

3.3

1/2

11:

1771561*T^4 - 23958*T^3 + 1870*T^2 - 18*T + 1

3.3

1/2

13:

4826809*T^4 - 114244*T^3 + 4017*T^2 - 52*T + 1

3.3

1/2

17:

24137569*T^4 - 176868*T^3 - 2567*T^2 - 36*T + 1

3.3

1/2

19:

47045881*T^4 + 260642*T^3 + 7410*T^2 + 38*T + 1

3.3

1/2

23:

148035889*T^4 + 1204533*T^3 + 11500*T^2 + 99*T + 1

3.3

1/2

29:

594823321*T^4 - 4390020*T^3 + 33379*T^2 - 180*T + 1

3.3

1/2

31:

887503681*T^4 + 8460644*T^3 + 59334*T^2 + 284*T + 1

3.3

-1

15625*T^4 + 375*T^3 + 70*T^2 + 3*T + 1

3.3

-1

117649*T^4 - 7546*T^3 + 483*T^2 - 22*T + 1

3.3

-1

11:

1771561*T^4 + 7986*T^3 - 1298*T^2 + 6*T + 1

3.3

-1

13:

4826809*T^4 + 10985*T^3 + 1950*T^2 + 5*T + 1

3.3

-1

17:

24137569*T^4 - 309519*T^3 + 1564*T^2 - 63*T + 1

3.3

-1

19:

47045881*T^4 + 96026*T^3 - 6726*T^2 + 14*T + 1

3.3

-1

23:

148035889*T^4 + 1022028*T^3 + 21643*T^2 + 84*T + 1

3.3

-1

29:

594823321*T^4 - 3585183*T^3 + 7018*T^2 - 147*T + 1

3.3

-1

31:

887503681*T^4 + 2025788*T^3 + 32550*T^2 + 68*T + 1

3.3

15625*T^4 + 750*T^3 - 20*T^2 + 6*T + 1

3.3

117649*T^4 + 4802*T^3 + 546*T^2 + 14*T + 1

3.3

11:

1771561*T^4 - 79860*T^3 + 2068*T^2 - 60*T + 1

3.3

13:

4826809*T^4 - 206518*T^3 + 4524*T^2 - 94*T + 1

3.3

17:

24137569*T^4 + 353736*T^3 + 9826*T^2 + 72*T + 1

3.3

19:

47045881*T^4 - 1015132*T^3 + 13794*T^2 - 148*T + 1

3.3

23:

148035889*T^4 + 1277535*T^3 + 4462*T^2 + 105*T + 1

3.3

29:

594823321*T^4 + 1609674*T^3 - 9164*T^2 + 66*T + 1

3.3

31:

887503681*T^4 - 119164*T^3 - 24087*T^2 - 4*T + 1

3.4

1/2

15625*T^4 + 1500*T^3 + 70*T^2 + 12*T + 1

3.4

1/2

117649*T^4 - 4802*T^3 + 266*T^2 - 14*T + 1

3.4

1/2

11:

1771561*T^4 + 95832*T^3 + 3454*T^2 + 72*T + 1

3.4

1/2

13:

4826809*T^4 - 4394*T^3 + 962*T^2 - 2*T + 1

3.4

1/2

17:

24137569*T^4 - 1190*T^2 + 1

3.4

1/2

19:

47045881*T^4 + 329232*T^3 - 1634*T^2 + 48*T + 1

3.4

1/2

23:

148035889*T^4 + 1168032*T^3 + 1150*T^2 + 96*T + 1

3.4

1/2

29:

594823321*T^4 + 10243380*T^3 + 89494*T^2 + 420*T + 1

3.4

1/2

31:

887503681*T^4 - 417074*T^3 + 5642*T^2 - 14*T + 1

3.4

-1

15625*T^4 - 875*T^3 + 70*T^2 - 7*T + 1

3.4

-1

117649*T^4 + 3773*T^3 - 42*T^2 + 11*T + 1

3.4

-1

11:

1771561*T^4 + 1331*T^3 - 506*T^2 + T + 1

3.4

-1

13:

4826809*T^4 - 26364*T^3 + 598*T^2 - 12*T + 1

3.4

-1

17:

24137569*T^4 + 9826*T^3 - 7922*T^2 + 2*T + 1

3.4

-1

19:

47045881*T^4 - 82308*T^3 + 1102*T^2 - 12*T + 1

3.4

-1

23:

148035889*T^4 + 1557376*T^3 + 20194*T^2 + 128*T + 1

3.4

-1

29:

594823321*T^4 + 2195010*T^3 + 13282*T^2 + 90*T + 1

3.4

-1

31:

887503681*T^4 + 387283*T^3 - 4402*T^2 + 13*T + 1

3.4

15625*T^4 + 125*T^3 - 20*T^2 + T + 1

3.4

117649*T^4 + 1029*T^3 - 266*T^2 + 3*T + 1

3.4

11:

1771561*T^4 - 38599*T^3 + 286*T^2 - 29*T + 1

3.4

13:

4826809*T^4 + 4394*T^3 + 1326*T^2 + 2*T + 1

3.4

17:

24137569*T^4 - 225998*T^3 + 5542*T^2 - 46*T + 1

3.4

19:

47045881*T^4 + 370386*T^3 - 1178*T^2 + 54*T + 1

3.4

23:

148035889*T^4 - 2068390*T^3 + 19366*T^2 - 170*T + 1

3.4

29:

594823321*T^4 + 1756008*T^3 - 10730*T^2 + 72*T + 1

3.4

31:

887503681*T^4 + 387283*T^3 + 49166*T^2 + 13*T + 1

3.6

1/2

15625*T^4 + 500*T^3 + 25*T^2 + 4*T + 1

3.6

1/2

117649*T^4 + 2401*T^3 - 28*T^2 + 7*T + 1

3.6

1/2

11:

1771561*T^4 - 31944*T^3 + 682*T^2 - 24*T + 1

3.6

1/2

13:

4826809*T^4 + 140608*T^3 + 3107*T^2 + 64*T + 1

3.6

1/2

17:

24137569*T^4 - 157216*T^3 - 119*T^2 - 32*T + 1

3.6

1/2

19:

47045881*T^4 + 109744*T^3 - 7030*T^2 + 16*T + 1

3.6

1/2

23:

148035889*T^4 - 474513*T^3 - 12512*T^2 - 39*T + 1

3.6

1/2

29:

594823321*T^4 + 2536456*T^3 + 47473*T^2 + 104*T + 1

3.6

1/2

31:

887503681*T^4 + 2621608*T^3 + 28334*T^2 + 88*T + 1

3.6

-1

15625*T^4 - 125*T^3 - 110*T^2 - T + 1

3.6

-1

117649*T^4 + 1372*T^3 + 203*T^2 + 4*T + 1

3.6

-1

11:

1771561*T^4 + 15972*T^3 + 682*T^2 + 12*T + 1

3.6

-1

13:

4826809*T^4 - 10985*T^3 - 1066*T^2 - 5*T + 1

3.6

-1

17:

24137569*T^4 + 34391*T^3 + 7684*T^2 + 7*T + 1

3.6

-1

19:

47045881*T^4 + 356668*T^3 + 1862*T^2 + 52*T + 1

3.6

-1

23:

148035889*T^4 - 1606044*T^3 + 24127*T^2 - 132*T + 1

3.6

-1

29:

594823321*T^4 - 1146283*T^3 + 15370*T^2 - 47*T + 1

3.6

-1

31:

887503681*T^4 + 476656*T^3 - 37510*T^2 + 16*T + 1

3.6

15625*T^4 - 2000*T^3 + 160*T^2 - 16*T + 1

3.6

117649*T^4 + 3430*T^3 + 518*T^2 + 10*T + 1

3.6

11:

1771561*T^4 - 79860*T^3 + 2068*T^2 - 60*T + 1

3.6

13:

4826809*T^4 + 8788*T^3 + 1664*T^2 + 4*T + 1

3.6

17:

24137569*T^4 - 393040*T^3 + 1870*T^2 - 80*T + 1

3.6

19:

47045881*T^4 + 768208*T^3 + 15542*T^2 + 112*T + 1

3.6

23:

148035889*T^4 - 620517*T^3 + 21022*T^2 - 51*T + 1

3.6

29:

594823321*T^4 - 1951120*T^3 - 3944*T^2 - 80*T + 1

3.6

31:

887503681*T^4 - 10784342*T^3 + 77159*T^2 - 362*T + 1

4.4

1/2

729*T^4 - 108*T^3 + 30*T^2 - 4*T + 1

4.4

1/2

15625*T^4 - 1000*T^3 - 30*T^2 - 8*T + 1

4.4

1/2

117649*T^4 + 1372*T^3 + 238*T^2 + 4*T + 1

4.4

1/2

11:

1771561*T^4 - 37268*T^3 + 462*T^2 - 28*T + 1

4.4

1/2

13:

4826809*T^4 - 193336*T^3 + 5746*T^2 - 88*T + 1

4.4

1/2

17:

24137569*T^4 + 98260*T^3 + 4726*T^2 + 20*T + 1

4.4

1/2

19:

47045881*T^4 - 1234620*T^3 + 19038*T^2 - 180*T + 1

4.4

1/2

23:

148035889*T^4 - 243340*T^3 + 7406*T^2 - 20*T + 1

4.4

1/2

29:

594823321*T^4 + 585336*T^3 + 3538*T^2 + 24*T + 1

4.4

1/2

31:

887503681*T^4 - 1922*T^2 + 1

4.4

-1

729*T^4 - 108*T^3 + 30*T^2 - 4*T + 1

4.4

-1

15625*T^4 + 1000*T^3 + 70*T^2 + 8*T + 1

4.4

-1

117649*T^4 + 8232*T^3 + 350*T^2 + 24*T + 1

4.4

-1

11:

1771561*T^4 - 15972*T^3 + 2222*T^2 - 12*T + 1

4.4

-1

13:

4826809*T^4 + 158184*T^3 + 2678*T^2 + 72*T + 1

4.4

-1

17:

24137569*T^4 + 58956*T^3 - 442*T^2 + 12*T + 1

4.4

-1

19:

47045881*T^4 + 960260*T^3 + 12350*T^2 + 140*T + 1

4.4

-1

23:

148035889*T^4 - 2238728*T^3 + 23230*T^2 - 184*T + 1

4.4

-1

29:

594823321*T^4 + 2536456*T^3 + 25462*T^2 + 104*T + 1

4.4

-1

31:

887503681*T^4 - 2859936*T^3 + 19902*T^2 - 96*T + 1

4.4

729*T^4 - 108*T^3 + 30*T^2 - 4*T + 1

4.4

15625*T^4 - 500*T^3 + 230*T^2 - 4*T + 1

4.4

117649*T^4 + 4116*T^3 + 238*T^2 + 12*T + 1

4.4

11:

1771561*T^4 + 47916*T^3 + 1870*T^2 + 36*T + 1

4.4

13:

4826809*T^4 - 8788*T^3 + 598*T^2 - 4*T + 1

4.4

17:

24137569*T^4 + 58956*T^3 + 374*T^2 + 12*T + 1

4.4

19:

47045881*T^4 + 82308*T^3 - 418*T^2 + 12*T + 1

4.4

23:

148035889*T^4 - 2482068*T^3 + 29486*T^2 - 204*T + 1

4.4

29:

594823321*T^4 + 3414460*T^3 + 27318*T^2 + 140*T + 1

4.4

31:

887503681*T^4 + 1429968*T^3 + 49662*T^2 + 48*T + 1

4.6

1/2

15625*T^4 + 500*T^3 + 150*T^2 + 4*T + 1

4.6

1/2

117649*T^4 - 3430*T^3 + 42*T^2 - 10*T + 1

4.6

1/2

11:

1771561*T^4 - 53240*T^3 + 638*T^2 - 40*T + 1

4.6

1/2

13:

4826809*T^4 - 65910*T^3 + 3874*T^2 - 30*T + 1

4.6

1/2

17:

24137569*T^4 + 707472*T^3 + 12954*T^2 + 144*T + 1

4.6

1/2

19:

47045881*T^4 + 219488*T^3 + 6878*T^2 + 32*T + 1

4.6

1/2

23:

148035889*T^4 + 778688*T^3 + 8510*T^2 + 64*T + 1

4.6

1/2

29:

594823321*T^4 + 292668*T^3 - 2842*T^2 + 12*T + 1

4.6

1/2

31:

887503681*T^4 + 9235210*T^3 + 60202*T^2 + 310*T + 1

4.6

-1

15625*T^4 - 125*T^3 - 10*T^2 - T + 1

4.6

-1

117649*T^4 + 1029*T^3 - 42*T^2 + 3*T + 1

4.6

-1

11:

1771561*T^4 - 43923*T^3 + 1606*T^2 - 33*T + 1

4.6

-1

13:

4826809*T^4 + 131820*T^3 + 4550*T^2 + 60*T + 1

4.6

-1

17:

24137569*T^4 + 186694*T^3 + 7310*T^2 + 38*T + 1

4.6

-1

19:

47045881*T^4 + 631028*T^3 + 11438*T^2 + 92*T + 1

4.6

-1

23:

148035889*T^4 - 97336*T^3 - 5566*T^2 - 8*T + 1

4.6

-1

29:

594823321*T^4 + 5609470*T^3 + 55042*T^2 + 230*T + 1

4.6

-1

31:

887503681*T^4 - 1757669*T^3 + 57102*T^2 - 59*T + 1

4.6

15625*T^4 + 1125*T^3 + 60*T^2 + 9*T + 1

4.6

117649*T^4 + 1715*T^3 + 630*T^2 + 5*T + 1

4.6

11:

1771561*T^4 - 3993*T^3 - 418*T^2 - 3*T + 1

4.6

13:

4826809*T^4 + 13182*T^3 + 2574*T^2 + 6*T + 1

4.6

17:

24137569*T^4 - 520778*T^3 + 9894*T^2 - 106*T + 1

4.6

19:

47045881*T^4 + 41154*T^3 + 3686*T^2 + 6*T + 1

4.6

23:

148035889*T^4 + 657018*T^3 - 4186*T^2 + 54*T + 1

4.6

29:

594823321*T^4 + 2146232*T^3 - 15370*T^2 + 88*T + 1

4.6

31:

887503681*T^4 - 3962203*T^3 + 15438*T^2 - 133*T + 1

6.6

1/2

15625*T^4 - 750*T^3 + 55*T^2 - 6*T + 1

6.6

1/2

117649*T^4 + 8575*T^3 + 532*T^2 + 25*T + 1

6.6

1/2

11:

1771561*T^4 - 18634*T^3 + 110*T^2 - 14*T + 1

6.6

1/2

13:

4826809*T^4 - 61516*T^3 + 481*T^2 - 28*T + 1

6.6

1/2

17:

24137569*T^4 + 530604*T^3 + 7769*T^2 + 108*T + 1

6.6

1/2

19:

47045881*T^4 - 41154*T^3 + 5586*T^2 - 6*T + 1

6.6

1/2

23:

148035889*T^4 + 2761909*T^3 + 29900*T^2 + 227*T + 1

6.6

1/2

29:

594823321*T^4 + 2438900*T^3 + 435*T^2 + 100*T + 1

6.6

1/2

31:

887503681*T^4 - 1549132*T^3 + 50406*T^2 - 52*T + 1

6.6

-1

15625*T^4 + 2375*T^3 + 230*T^2 + 19*T + 1

6.6

-1

117649*T^4 - 6174*T^3 + 595*T^2 - 18*T + 1

6.6

-1

11:

1771561*T^4 - 39930*T^3 + 1166*T^2 - 30*T + 1

6.6

-1

13:

4826809*T^4 - 59319*T^3 + 1118*T^2 - 27*T + 1

6.6

-1

17:

24137569*T^4 + 162129*T^3 - 612*T^2 + 33*T + 1

6.6

-1

19:

47045881*T^4 + 288078*T^3 + 4826*T^2 + 42*T + 1

6.6

-1

23:

148035889*T^4 - 146004*T^3 - 11845*T^2 - 12*T + 1

6.6

-1

29:

594823321*T^4 + 10853105*T^3 + 91466*T^2 + 445*T + 1

6.6

-1

31:

887503681*T^4 - 1072476*T^3 + 10726*T^2 - 36*T + 1

6.6

15625*T^4 - 750*T^3 - 20*T^2 - 6*T + 1

6.6

117649*T^4 - 686*T^3 - 350*T^2 - 2*T + 1

6.6

11:

1771561*T^4 + 90508*T^3 + 3476*T^2 + 68*T + 1

6.6

13:

4826809*T^4 - 162578*T^3 + 4108*T^2 - 74*T + 1

6.6

17:

24137569*T^4 + 353736*T^3 + 3842*T^2 + 72*T + 1

6.6

19:

47045881*T^4 + 246924*T^3 + 2850*T^2 + 36*T + 1

6.6

23:

148035889*T^4 + 498847*T^3 + 5198*T^2 + 41*T + 1

6.6

29:

594823321*T^4 + 2682790*T^3 + 34452*T^2 + 110*T + 1

6.6

31:

887503681*T^4 + 1310804*T^3 - 13671*T^2 + 44*T + 1

1/2

729*T^4 + 135*T^3 + 45*T^2 + 5*T + 1

1/2

117649*T^4 - 12005*T^3 + 805*T^2 - 35*T + 1

1/2

11:

1771561*T^4 - 11979*T^3 - 44*T^2 - 9*T + 1

1/2

13:

4826809*T^4 + 21970*T^3 - 910*T^2 + 10*T + 1

1/2

17:

24137569*T^4 - 687820*T^3 + 12070*T^2 - 140*T + 1

1/2

19:

47045881*T^4 - 754490*T^3 + 8018*T^2 - 110*T + 1

1/2

23:

148035889*T^4 + 1277535*T^3 + 8395*T^2 + 105*T + 1

1/2

29:

594823321*T^4 - 5975305*T^3 + 50228*T^2 - 245*T + 1

1/2

31:

887503681*T^4 - 4587814*T^3 + 5766*T^2 - 154*T + 1

-1

729*T^4 + 135*T^3 + 45*T^2 + 5*T + 1

-1

117649*T^4 + 8575*T^3 + 350*T^2 + 25*T + 1

-1

11:

1771561*T^4 - 18634*T^3 + 836*T^2 - 14*T + 1

-1

13:

4826809*T^4 - 32955*T^3 - 260*T^2 - 15*T + 1

-1

17:

24137569*T^4 + 73695*T^3 + 1530*T^2 + 15*T + 1

-1

19:

47045881*T^4 - 823080*T^3 + 9918*T^2 - 120*T + 1

-1

23:

148035889*T^4 - 730020*T^3 + 5060*T^2 - 60*T + 1

-1

29:

594823321*T^4 + 1097505*T^3 + 13978*T^2 + 45*T + 1

-1

31:

887503681*T^4 - 7715869*T^3 + 49321*T^2 - 259*T + 1

729*T^4 + 135*T^3 + 45*T^2 + 5*T + 1

117649*T^4 + 3430*T^3 + 420*T^2 + 10*T + 1

11:

1771561*T^4 - 18634*T^3 - 814*T^2 - 14*T + 1

13:

4826809*T^4 + 186745*T^3 + 5265*T^2 + 85*T + 1

17:

24137569*T^4 + 98260*T^3 + 2040*T^2 + 20*T + 1

19:

47045881*T^4 + 137180*T^3 + 418*T^2 + 20*T + 1

23:

148035889*T^4 + 608350*T^3 + 10350*T^2 + 50*T + 1

29:

594823321*T^4 - 2195010*T^3 - 3422*T^2 - 90*T + 1

31:

887503681*T^4 + 5690081*T^3 + 33821*T^2 + 191*T + 1

1/2

729*T^4 - 108*T^3 + 18*T^2 - 4*T + 1

1/2

15625*T^4 - 2000*T^3 + 300*T^2 - 16*T + 1

1/2

117649*T^4 + 2058*T^3 + 70*T^2 + 6*T + 1

1/2

11:

1771561*T^4 + 15972*T^3 + 946*T^2 + 12*T + 1

1/2

13:

4826809*T^4 - 17576*T^3 + 156*T^2 - 8*T + 1

1/2

17:

24137569*T^4 + 39304*T^3 + 4590*T^2 + 8*T + 1

1/2

19:

47045881*T^4 - 466412*T^3 + 6346*T^2 - 68*T + 1

1/2

23:

148035889*T^4 + 2068390*T^3 + 21390*T^2 + 170*T + 1

1/2

29:

594823321*T^4 - 3902240*T^3 + 29116*T^2 - 160*T + 1

1/2

31:

887503681*T^4 + 2740772*T^3 - 4898*T^2 + 92*T + 1

-1

729*T^4 - 108*T^3 + 18*T^2 - 4*T + 1

-1

15625*T^4 + 1750*T^3 + 90*T^2 + 14*T + 1

-1

117649*T^4 - 2744*T^3 - 350*T^2 - 8*T + 1

-1

11:

1771561*T^4 - 79860*T^3 + 2794*T^2 - 60*T + 1

-1

13:

4826809*T^4 - 57122*T^3 + 3666*T^2 - 26*T + 1

-1

17:

24137569*T^4 - 373388*T^3 + 5814*T^2 - 76*T + 1

-1

19:

47045881*T^4 - 795644*T^3 + 9234*T^2 - 116*T + 1

-1

23:

148035889*T^4 + 292008*T^3 + 4738*T^2 + 24*T + 1

-1

29:

594823321*T^4 + 731670*T^3 + 17690*T^2 + 30*T + 1

-1

31:

887503681*T^4 - 4766560*T^3 + 40238*T^2 - 160*T + 1

729*T^4 - 108*T^3 + 18*T^2 - 4*T + 1

15625*T^4 + 500*T^3 + 10*T^2 + 4*T + 1

117649*T^4 + 2058*T^3 + 238*T^2 + 6*T + 1

11:

1771561*T^4 - 5324*T^3 + 1650*T^2 - 4*T + 1

13:

4826809*T^4 - 149396*T^3 + 4498*T^2 - 68*T + 1

17:

24137569*T^4 + 530604*T^3 + 10914*T^2 + 108*T + 1

19:

47045881*T^4 + 82308*T^3 + 3306*T^2 + 12*T + 1

23:

148035889*T^4 + 170338*T^3 - 9522*T^2 + 14*T + 1

29:

594823321*T^4 - 8097148*T^3 + 62466*T^2 - 332*T + 1

31:

887503681*T^4 - 8103152*T^3 + 46190*T^2 - 272*T + 1

1/2

729*T^4 + 135*T^3 + 21*T^2 + 5*T + 1

1/2

117649*T^4 - 9947*T^3 + 413*T^2 - 29*T + 1

1/2

11:

1771561*T^4 + 11979*T^3 - 44*T^2 + 9*T + 1

1/2

13:

4826809*T^4 + 57122*T^3 + 1794*T^2 + 26*T + 1

1/2

17:

24137569*T^4 - 569908*T^3 + 12070*T^2 - 116*T + 1

1/2

19:

47045881*T^4 - 342950*T^3 + 1634*T^2 - 50*T + 1

1/2

23:

148035889*T^4 + 863857*T^3 + 1587*T^2 + 71*T + 1

1/2

29:

594823321*T^4 - 463391*T^3 - 24476*T^2 - 19*T + 1

1/2

31:

887503681*T^4 + 178746*T^3 - 10106*T^2 + 6*T + 1

-1

729*T^4 + 135*T^3 + 21*T^2 + 5*T + 1

-1

117649*T^4 - 5145*T^3 + 238*T^2 - 15*T + 1

-1

11:

1771561*T^4 - 13310*T^3 - 44*T^2 - 10*T + 1

-1

13:

4826809*T^4 - 177957*T^3 + 5148*T^2 - 81*T + 1

-1

17:

24137569*T^4 - 545343*T^3 + 9418*T^2 - 111*T + 1

-1

19:

47045881*T^4 + 274360*T^3 + 5662*T^2 + 40*T + 1

-1

23:

148035889*T^4 + 1216700*T^3 + 20884*T^2 + 100*T + 1

-1

29:

594823321*T^4 + 121945*T^3 + 7946*T^2 + 5*T + 1

-1

31:

887503681*T^4 - 2532235*T^3 + 29233*T^2 - 85*T + 1

729*T^4 + 135*T^3 + 21*T^2 + 5*T + 1

117649*T^4 + 2058*T^3 - 252*T^2 + 6*T + 1

11:

1771561*T^4 + 7986*T^3 + 1826*T^2 + 6*T + 1

13:

4826809*T^4 + 63713*T^3 - 1391*T^2 + 29*T + 1

17:

24137569*T^4 - 176868*T^3 + 8840*T^2 - 36*T + 1

19:

47045881*T^4 - 960260*T^3 + 15314*T^2 - 140*T + 1

23:

148035889*T^4 - 900358*T^3 + 15502*T^2 - 74*T + 1

29:

594823321*T^4 - 1121894*T^3 + 19314*T^2 - 46*T + 1

31:

887503681*T^4 - 2651399*T^3 + 4309*T^2 - 89*T + 1

1/2

15625*T^4 + 500*T^3 + 135*T^2 + 4*T + 1

1/2

117649*T^4 + 4459*T^3 + 364*T^2 + 13*T + 1

1/2

11:

1771561*T^4 + 42592*T^3 + 2046*T^2 + 32*T + 1

1/2

13:

4826809*T^4 + 17576*T^3 + 3757*T^2 + 8*T + 1

1/2

17:

24137569*T^4 - 196520*T^3 + 5185*T^2 - 40*T + 1

1/2

19:

47045881*T^4 - 9006*T^2 + 1

1/2

23:

148035889*T^4 - 1666879*T^3 + 8924*T^2 - 137*T + 1

1/2

29:

594823321*T^4 - 585336*T^3 - 12905*T^2 - 24*T + 1

1/2

31:

887503681*T^4 - 7149840*T^3 + 29326*T^2 - 240*T + 1

-1

15625*T^4 - 125*T^3 + 190*T^2 - T + 1

-1

117649*T^4 - 12348*T^3 + 875*T^2 - 36*T + 1

-1

11:

1771561*T^4 - 15972*T^3 - 946*T^2 - 12*T + 1

-1

13:

4826809*T^4 + 72501*T^3 + 3302*T^2 + 33*T + 1

-1

17:

24137569*T^4 + 44217*T^3 + 476*T^2 + 9*T + 1

-1

19:

47045881*T^4 + 631028*T^3 + 4370*T^2 + 92*T + 1

-1

23:

148035889*T^4 - 340676*T^3 - 6785*T^2 - 28*T + 1

-1

29:

594823321*T^4 + 5389969*T^3 + 43210*T^2 + 221*T + 1

-1

31:

887503681*T^4 - 8579808*T^3 + 60450*T^2 - 288*T + 1

15625*T^4 - 2000*T^3 + 140*T^2 - 16*T + 1

117649*T^4 + 2058*T^3 + 238*T^2 + 6*T + 1

11:

1771561*T^4 - 15972*T^3 + 748*T^2 - 12*T + 1

13:

4826809*T^4 - 8788*T^3 + 2236*T^2 - 4*T + 1

17:

24137569*T^4 - 353736*T^3 + 4862*T^2 - 72*T + 1

19:

47045881*T^4 - 1152312*T^3 + 16682*T^2 - 168*T + 1

23:

148035889*T^4 - 474513*T^3 + 12374*T^2 - 39*T + 1

29:

594823321*T^4 + 1365784*T^3 + 1740*T^2 + 56*T + 1

31:

887503681*T^4 + 714984*T^3 - 39339*T^2 + 24*T + 1

Definition

The polynomial $P_{A,z,p}(T)=\det(1-T\operatorname{Frob}_p\mid H_{A,z})$ of geometric Frobenius at a good rational prime $p$ on the rank-four, weight-three hypergeometric Calabi–Yau motive $H_{A,z}$, with Hodge vector $(1,1,1,1)$ and no additional Tate twist [1] [2].

Parameters

$A$

— hypergeometric template ($A$ is one of the fourteen cyclotomic numerator multisets specified in (4))

$z$

— rational fibre ($z\in\mathbb Q\setminus\{0,1\}$)

$p$

— good rational prime ($p$ is prime and $p\nmid d u v(u-v)$, where $z=u/v$ is reduced and $d$ is the least common multiple of the indices in $A$)

Formulas

(1)

If $a_p=\operatorname{Tr}(\operatorname{Frob}_p\mid H_{A,z})$ and $c_{p,2}=\operatorname{Tr}(\operatorname{Frob}_p^2\mid H_{A,z})$, then $b_p=(a_p^2-c_{p,2})/2$ and $P_{A,z,p}(T)=1-a_pT+b_pT^2-p^3a_pT^3+p^6T^4$ [2].

(2)

$Q_{A,z,p}(X)=X^4P_{A,z,p}(1/X)=X^4-a_pX^3+b_pX^2-p^3a_pX+p^6$ is the monic polynomial of the Frobenius eigenvalues.

(3)

$p^6T^4P_{A,z,p}(1/(p^3T))=P_{A,z,p}(T)$ [2].

Comments

(4)

The template $A=(d_1,\ldots,d_k)$ denotes the cyclotomic quotient $\prod_j\Phi_{d_j}/\Phi_1^4$. Its upper multiset consists of all $a/d_j$ with $1\leq a<d_j$ and $\gcd(a,d_j)=1$, including multiplicities; its lower multiset is $(0,0,0,0)$ modulo integers. The fourteen keys are $2.2.2.2$, $2.2.3$, $2.2.4$, $2.2.6$, $3.3$, $3.4$, $3.6$, $4.4$, $4.6$, $6.6$, $5$, $8$, $10$ and $12$. Dots separate cyclotomic indices, not decimal places. This is the complete template class in Table 1 and Section 5 of [1].

(5)

The coordinate $z$ is the classical argument of ${}_4F_3(\alpha;1,1,1;z)$, with singular points $0$, $1$ and $\infty$. It agrees with Sage [4]; PARI and Magma use $t=1/z$ [5]. A mirror parameter or a rescaled factorial-series coordinate is not substituted for $z$.

(6)

The motive is the rank-four hypergeometric summand, rather than the entire third cohomology of every geometric model. The singular fibre $z=1$ requires a separate treatment [1].

(7)

The Frobenius eigenvalues have complex absolute value $p^{3/2}$. The constant-term-one convention agrees with the zeta polynomials of Fermat curves. The Weil polynomials of abelian varieties instead use the monic eigenvalue-polynomial convention; those motives have weight one.

(8)

Distinct rational fibres may reduce to the same element of $\mathbb F_p$ and then have the same local factor. Their global-fibre identities are retained. Reducible factors and zero traces are included.

(9)

Finite hypergeometric traces are assembled from Gauss sums [3]. The Gauss-sum table and the Jacobi-sum table give prime-field character sums related to these constructions; they do not store the resulting rank-four local factors.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.symbolic.ring import SR
from sage.modular.hypergeometric_motive import HypergeometricData
H = HypergeometricData(cyclotomic=([5], [1,1,1,1]))
H.euler_factor(QQ(1)/2, ZZ(7))

(P2)

PARI/GP

H = hgminit([5]);
z = 1/2; p = 7;
hgmeulerfactor(H, 1/z, p)

References

[1]

Ling Long, Fang-Ting Tu, Noriko Yui and Wadim Zudilin, Supercongruences for rigid hypergeometric Calabi–Yau threefolds, Table 1 and Section 5. (arXiv)

[2]

David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, Sections 10–11. (arXiv)

[3]

Frits Beukers, Henri Cohen and Anton Mellit, Finite hypergeometric functions, Pure and Applied Mathematics Quarterly 11 (2015), 559–589, Definition 1.1 and Theorems 1.2–1.3. (arXiv)

Links

[4]

SageMath: Hypergeometric motives

[5]

PARI/GP: Hypergeometric Motives

Similar tables

Zeta_polynomials_of_Fermat_curves — uses the same constant-term-one convention for local factors of weight-one curve cohomology

Weil_polynomials_of_abelian_varieties_over_prime_fields — uses the monic reciprocal convention and indexes finite-field isogeny classes

Gauss_sums_of_primitive_Dirichlet_characters — gives individual Gauss sums of primitive Dirichlet characters

Jacobi_sums_of_pairs_of_Dirichlet_characters_modulo_a_prime — gives prime-field Jacobi sums related to finite hypergeometric trace formulas

Unit roots of hypergeometric Calabi–Yau motives — tabulates the unique $p$-adic unit eigenvalue $u_{A,z,p}$ for the same $(A,z,p)$ when $p\nmid a_p$. It satisfies $Q_{A,z,p}(u_{A,z,p})=0$ for $Q(X)=X^4P(1/X)$, so its reciprocal is a zero of the local factor stored here. The factors with $p\mid a_p$ have no unit-root entry

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds all fourteen templates at every rational solution of $z,1-z\in\mathbb Z[1/2]^\times$, namely $z=1/2,-1,2$, and every good prime $p\leq31$, giving $390$ local factors. The fibre restriction introduces no potentially bad prime outside $2$ beyond the template wild primes. The allowed extra-prime set $\{2\}$ and prime cutoff are noncanonical coverage choices: all templates receive several reductions, including different splitting classes, before increasing prime depth. The attached argument-selection note explains the arithmetic selection and practical extent)

How they were obtained:

Exact integer polynomials reconstructed from Sage Gross–Koblitz traces over $\mathbb F_p$ and $\mathbb F_{p^2}$. For each extension degree $f$, the modulus $p^N$ satisfies $p^N>8p^{3f/2}$; the Weil bound $|\operatorname{Tr}(\operatorname{Frob}_p^f)|\leq4p^{3f/2}$ then singles out the centered integer. Recomputations with three additional $p$-adic digits agree.

Every full polynomial agrees with PARI at $1/z$ and with a separate implementation of the finite-field Gauss-sum formula in Theorem 1.3 of [3]. The explicit Gauss-sum computation uses 384-bit complex balls, requires finite real and imaginary radii smaller than $10^{-75}$, and isolates a unique integer using the integrality of the trace. Nonzero Gauss-norm and published-polynomial controls precede the comparisons. Eighteen quadratic-character convolutions and Legendre elliptic point counts check another arithmetic route, including the quadratic-character normalization. Degree, constant and leading coefficients, reciprocity, Newton identities and exact Weil-root bounds are checked throughout. The attached scripts apply these checks both offline and to the values read from the draft; numerical evidence records the actual runs.