Weber class polynomials $W_D$

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Polynomials

$D$

$W_D(x)$

-7:

x - 1

comment: $h(D)=1$; fundamental discriminant of $\mathbb{Q}(\sqrt{-7})$

-23:

x^3 - x - 1

comment: $h(D)=3$; fundamental discriminant of $\mathbb{Q}(\sqrt{-23})$

-31:

x^3 - x^2 - 1

comment: $h(D)=3$; fundamental discriminant of $\mathbb{Q}(\sqrt{-31})$

-47:

x^5 - x^3 - 2*x^2 - 2*x - 1

comment: $h(D)=5$; fundamental discriminant of $\mathbb{Q}(\sqrt{-47})$

-55:

x^4 - 2*x^3 + x - 1

comment: $h(D)=4$; fundamental discriminant of $\mathbb{Q}(\sqrt{-55})$

-71:

x^7 - 2*x^6 - x^5 + x^4 + x^3 + x^2 - x - 1

comment: $h(D)=7$; fundamental discriminant of $\mathbb{Q}(\sqrt{-71})$

-79:

x^5 - 3*x^4 + 2*x^3 - x^2 + x - 1

comment: $h(D)=5$; fundamental discriminant of $\mathbb{Q}(\sqrt{-79})$

-95:

x^8 - 2*x^7 - 2*x^6 + x^5 + 2*x^4 - x^3 + x - 1

comment: $h(D)=8$; fundamental discriminant of $\mathbb{Q}(\sqrt{-95})$

-103:

x^5 - x^4 - 3*x^3 - 3*x^2 - 2*x - 1

comment: $h(D)=5$; fundamental discriminant of $\mathbb{Q}(\sqrt{-103})$

-119:

x^10 - 4*x^9 + 5*x^8 - 8*x^7 + 9*x^6 - 7*x^5 + 5*x^4 - 4*x^3 + 2*x^2 - x + 1

comment: $h(D)=10$; fundamental discriminant of $\mathbb{Q}(\sqrt{-119})$

-127:

x^5 - 3*x^4 - x^3 + 2*x^2 + x - 1

comment: $h(D)=5$; fundamental discriminant of $\mathbb{Q}(\sqrt{-127})$

-143:

x^10 - 6*x^9 + 12*x^8 - 13*x^7 + 9*x^6 - 3*x^5 - 3*x^4 + 6*x^3 - 6*x^2 + 3*x - 1

comment: $h(D)=10$; fundamental discriminant of $\mathbb{Q}(\sqrt{-143})$

-151:

x^7 - 3*x^6 - x^5 - 3*x^4 - x^2 - x - 1

comment: $h(D)=7$; fundamental discriminant of $\mathbb{Q}(\sqrt{-151})$

-167:

x^11 - 2*x^10 - 4*x^9 - 9*x^8 - 7*x^7 - 11*x^6 - 6*x^5 - 10*x^4 - 4*x^3 - 5*x^2 - x - 1

comment: $h(D)=11$; fundamental discriminant of $\mathbb{Q}(\sqrt{-167})$

-175:

x^6 - 4*x^5 + x + 1

comment: $h(D)=6$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-7})$

-191:

x^13 - 6*x^12 + 10*x^11 - 16*x^10 + 22*x^9 - 19*x^8 + 11*x^7 - 5*x^6 - x^5 + 5*x^4 - 4*x^3 + 2*x - 1

comment: $h(D)=13$; fundamental discriminant of $\mathbb{Q}(\sqrt{-191})$

-199:

x^9 - 5*x^8 + 3*x^7 - 3*x^6 - 3*x^3 - x - 1

comment: $h(D)=9$; fundamental discriminant of $\mathbb{Q}(\sqrt{-199})$

-215:

x^14 - 6*x^13 + 4*x^12 + 11*x^11 - 13*x^10 - 7*x^9 + 16*x^8 - 4*x^7 - 13*x^6 + 8*x^5 + 3*x^4 - 6*x^3 + 2*x - 1

comment: $h(D)=14$; fundamental discriminant of $\mathbb{Q}(\sqrt{-215})$

-223:

x^7 - 5*x^6 + x^4 - 4*x^3 - x^2 - 1

comment: $h(D)=7$; fundamental discriminant of $\mathbb{Q}(\sqrt{-223})$

-239:

x^15 - 6*x^14 + 2*x^13 + 8*x^12 + 4*x^11 - 27*x^10 + 13*x^9 + 15*x^8 - 4*x^7 - 20*x^6 + 13*x^5 + 5*x^4 - 4*x^3 - 4*x^2 + 4*x - 1

comment: $h(D)=15$; fundamental discriminant of $\mathbb{Q}(\sqrt{-239})$

-247:

x^6 - 4*x^5 - 7*x^4 - 7*x^3 - 6*x^2 - 3*x - 1

comment: $h(D)=6$; fundamental discriminant of $\mathbb{Q}(\sqrt{-247})$

-263:

x^13 - 8*x^12 + 16*x^11 - 27*x^10 + 38*x^9 - 36*x^8 + 22*x^7 - 12*x^6 + 13*x^5 - 19*x^4 + 21*x^3 - 15*x^2 + 6*x - 1

comment: $h(D)=13$; fundamental discriminant of $\mathbb{Q}(\sqrt{-263})$

-271:

x^11 - 5*x^10 - 6*x^9 - 5*x^8 + 3*x^7 + 6*x^6 + 3*x^5 - 3*x^4 - x^3 - x^2 - 1

comment: $h(D)=11$; fundamental discriminant of $\mathbb{Q}(\sqrt{-271})$

-287:

x^14 - 8*x^13 + 9*x^12 + 6*x^11 - 5*x^10 - 7*x^9 - 8*x^8 + 6*x^7 + 2*x^6 - x^4 - x^3 + 3*x^2 + x + 1

comment: $h(D)=14$; fundamental discriminant of $\mathbb{Q}(\sqrt{-287})$

-295:

x^8 - 8*x^7 + 9*x^6 - x^5 - 7*x^4 + 10*x^3 - 7*x^2 + 3*x - 1

comment: $h(D)=8$; fundamental discriminant of $\mathbb{Q}(\sqrt{-295})$

-311:

x^19 - 4*x^18 - 16*x^17 - 37*x^16 - 42*x^15 - 38*x^14 - 4*x^13 + 10*x^12 + 25*x^11 + 18*x^10 + 9*x^9 + x^8 - 10*x^7 - 13*x^6 - 14*x^5 - 8*x^4 - 5*x^3 - 2*x^2 - x - 1

comment: $h(D)=19$; fundamental discriminant of $\mathbb{Q}(\sqrt{-311})$

-319:

x^10 - 6*x^9 - 9*x^8 - 5*x^7 - x^6 - 2*x^5 - 10*x^4 - 14*x^3 - 11*x^2 - 5*x - 1

comment: $h(D)=10$; fundamental discriminant of $\mathbb{Q}(\sqrt{-319})$

-335:

x^18 - 4*x^17 - 20*x^16 - 55*x^15 - 106*x^14 - 144*x^13 - 163*x^12 - 174*x^11 - 179*x^10 - 171*x^9 - 144*x^8 - 102*x^7 - 64*x^6 - 42*x^5 - 33*x^4 - 25*x^3 - 14*x^2 - 5*x - 1

comment: $h(D)=18$; fundamental discriminant of $\mathbb{Q}(\sqrt{-335})$

-343:

x^7 - 7*x^6 - 7*x^5 - 7*x^4 - 1

comment: $h(D)=7$; order of conductor $7$ in $\mathbb{Q}(\sqrt{-7})$

-359:

x^19 - 14*x^18 + 59*x^17 - 113*x^16 + 91*x^15 + 19*x^14 - 90*x^13 + 51*x^12 + 2*x^11 - 5*x^10 + 9*x^9 - 30*x^8 + 22*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 - 2*x^2 + 2*x - 1

comment: $h(D)=19$; fundamental discriminant of $\mathbb{Q}(\sqrt{-359})$

-367:

x^9 - 9*x^8 + 3*x^7 - 2*x^6 + 2*x^4 - 6*x^3 + x^2 + 2*x - 1

comment: $h(D)=9$; fundamental discriminant of $\mathbb{Q}(\sqrt{-367})$

-383:

x^17 - 6*x^16 - 24*x^15 - 42*x^14 - 31*x^13 - 23*x^12 - 7*x^11 - x^10 - 4*x^9 - 11*x^8 - 7*x^7 - 13*x^6 - x^5 + x^3 + x^2 + x - 1

comment: $h(D)=17$; fundamental discriminant of $\mathbb{Q}(\sqrt{-383})$

-391:

x^14 - 8*x^13 - 12*x^12 - 12*x^11 + 4*x^9 - 3*x^8 - 8*x^7 + 6*x^6 + 15*x^5 + 7*x^4 - 5*x^3 - 5*x^2 + 1

comment: $h(D)=14$; fundamental discriminant of $\mathbb{Q}(\sqrt{-391})$

-407:

x^16 - 10*x^15 + 4*x^14 - 33*x^13 + 17*x^12 - 19*x^11 - 4*x^10 - 12*x^8 - 15*x^6 + 2*x^5 - 9*x^4 - x^3 - 2*x^2 - x - 1

comment: $h(D)=16$; fundamental discriminant of $\mathbb{Q}(\sqrt{-407})$

-415:

x^10 - 10*x^9 - x^8 - 7*x^7 - 11*x^6 + 2*x^5 - 14*x^4 + 2*x^3 - 6*x^2 - 1

comment: $h(D)=10$; fundamental discriminant of $\mathbb{Q}(\sqrt{-415})$

-431:

x^21 - 12*x^20 + 15*x^19 - 16*x^18 + 40*x^17 - 21*x^16 - 10*x^15 - 44*x^14 - 83*x^13 - 41*x^12 - 66*x^11 + 2*x^10 + 14*x^9 + 30*x^8 + 36*x^7 + 10*x^6 + 4*x^5 - 9*x^4 - 9*x^3 - 6*x^2 - 3*x - 1

comment: $h(D)=21$; fundamental discriminant of $\mathbb{Q}(\sqrt{-431})$

-439:

x^15 - 13*x^14 + 23*x^13 - 7*x^12 - 24*x^11 + 20*x^10 + 13*x^9 - 38*x^8 + 29*x^7 + x^6 - 17*x^5 + 7*x^4 + 9*x^3 - 11*x^2 + 5*x - 1

comment: $h(D)=15$; fundamental discriminant of $\mathbb{Q}(\sqrt{-439})$

-455:

x^20 - 6*x^19 - 50*x^18 - 142*x^17 - 200*x^16 - 129*x^15 + 38*x^14 + 191*x^13 + 246*x^12 + 194*x^11 + 76*x^10 - 30*x^9 - 73*x^8 - 57*x^7 - 15*x^6 + 16*x^5 + 26*x^4 + 23*x^3 + 15*x^2 + 6*x + 1

comment: $h(D)=20$; fundamental discriminant of $\mathbb{Q}(\sqrt{-455})$

-463:

x^7 - 11*x^6 - 9*x^5 - 8*x^4 - 7*x^3 - 7*x^2 - 3*x - 1

comment: $h(D)=7$; fundamental discriminant of $\mathbb{Q}(\sqrt{-463})$

-479:

x^25 - 10*x^24 - 23*x^23 - 76*x^22 - 104*x^21 - 126*x^20 - 109*x^19 - 144*x^18 - 205*x^17 - 317*x^16 - 336*x^15 - 280*x^14 - 138*x^13 - 11*x^12 + 76*x^11 + 82*x^10 + 48*x^9 - 6*x^8 - 47*x^7 - 66*x^6 - 60*x^5 - 41*x^4 - 22*x^3 - 9*x^2 - 3*x - 1

comment: $h(D)=25$; fundamental discriminant of $\mathbb{Q}(\sqrt{-479})$

-487:

x^7 - 13*x^6 + 4*x^5 - 4*x^4 + 7*x^3 - 4*x^2 + x - 1

comment: $h(D)=7$; fundamental discriminant of $\mathbb{Q}(\sqrt{-487})$

-503:

x^21 - 18*x^20 + 69*x^19 - 87*x^18 - 34*x^17 + 171*x^16 - 106*x^15 - 74*x^14 + 92*x^13 + 19*x^12 - 27*x^11 - 30*x^10 + 23*x^9 + 10*x^8 - 12*x^7 + 3*x^6 - 7*x^4 + 6*x^3 + x^2 - 1

comment: $h(D)=21$; fundamental discriminant of $\mathbb{Q}(\sqrt{-503})$

-511:

x^14 - 16*x^13 + 36*x^12 - 56*x^11 + 77*x^10 - 84*x^9 + 70*x^8 - 37*x^7 + 16*x^6 - 21*x^5 + 35*x^4 - 35*x^3 + 21*x^2 - 7*x + 1

comment: $h(D)=14$; fundamental discriminant of $\mathbb{Q}(\sqrt{-511})$

-527:

x^18 - 12*x^17 - 31*x^16 - 24*x^15 + 45*x^14 + 15*x^13 + 12*x^12 + 20*x^11 - 5*x^10 - 32*x^9 - 10*x^8 - x^7 - 7*x^6 - 4*x^5 + 8*x^4 + 5*x^3 + 2*x^2 + 1

comment: $h(D)=18$; fundamental discriminant of $\mathbb{Q}(\sqrt{-527})$

-535:

x^14 - 16*x^13 + 20*x^12 + 7*x^11 - 14*x^10 + 10*x^9 - 37*x^8 + 34*x^7 + 3*x^6 - x^5 - 16*x^4 + 6*x^3 + 3*x^2 - 1

comment: $h(D)=14$; fundamental discriminant of $\mathbb{Q}(\sqrt{-535})$

-551:

x^26 - 10*x^25 - 62*x^24 - 237*x^23 - 618*x^22 - 1183*x^21 - 1773*x^20 - 2121*x^19 - 2049*x^18 - 1625*x^17 - 1011*x^16 - 416*x^15 - 25*x^14 + 148*x^13 + 111*x^12 - 95*x^11 - 323*x^10 - 469*x^9 - 504*x^8 - 437*x^7 - 316*x^6 - 198*x^5 - 109*x^4 - 49*x^3 - 17*x^2 - 5*x - 1

comment: $h(D)=26$; fundamental discriminant of $\mathbb{Q}(\sqrt{-551})$

-559:

x^16 - 20*x^15 + 78*x^14 - 161*x^13 + 196*x^12 - 139*x^11 + 18*x^10 + 83*x^9 - 123*x^8 + 98*x^7 - 51*x^6 + 14*x^5 - x^4 - 2*x^3 + x - 1

comment: $h(D)=16$; fundamental discriminant of $\mathbb{Q}(\sqrt{-559})$

-575:

x^18 - 20*x^17 + 64*x^16 - 64*x^15 + 25*x^12 - 60*x^11 - x^8 - 15*x^7 + 6*x^6 + 4*x^5 + x^3 + 5*x^2 + 4*x + 1

comment: $h(D)=18$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-23})$

-583:

x^8 - 16*x^7 - 12*x^6 + 11*x^5 + 12*x^4 + 5*x^3 - 3*x^2 - 4*x - 1

comment: $h(D)=8$; fundamental discriminant of $\mathbb{Q}(\sqrt{-583})$

-599:

x^25 - 16*x^24 - 25*x^23 + 3*x^22 + 82*x^21 - 92*x^20 - 63*x^19 - 28*x^18 + 66*x^17 + 24*x^16 - 35*x^15 + 71*x^13 + 6*x^12 - 58*x^11 - 22*x^10 + 36*x^9 + 13*x^8 - 32*x^7 + 5*x^6 + 17*x^5 - 6*x^4 - 8*x^3 + x^2 + 2*x - 1

comment: $h(D)=25$; fundamental discriminant of $\mathbb{Q}(\sqrt{-599})$

-607:

x^13 - 17*x^12 - 15*x^11 + 16*x^10 + 34*x^9 + 3*x^8 - 21*x^7 - 21*x^6 + 9*x^5 + 7*x^4 + x^3 - 3*x^2 + 2*x - 1

comment: $h(D)=13$; fundamental discriminant of $\mathbb{Q}(\sqrt{-607})$

-623:

x^22 - 22*x^21 + 73*x^20 - 190*x^19 + 442*x^18 - 708*x^17 + 787*x^16 - 712*x^15 + 546*x^14 - 303*x^13 + 129*x^12 - 74*x^11 + 39*x^10 + 3*x^9 - 38*x^8 + 61*x^7 - 57*x^6 + 34*x^5 - 7*x^4 - 12*x^3 + 14*x^2 - 6*x + 1

comment: $h(D)=22$; fundamental discriminant of $\mathbb{Q}(\sqrt{-623})$

-631:

x^13 - 17*x^12 - 33*x^11 - 69*x^10 - 77*x^9 - 71*x^8 - 72*x^7 - 54*x^6 - 46*x^5 - 37*x^4 - 21*x^3 - 12*x^2 - 4*x - 1

comment: $h(D)=13$; fundamental discriminant of $\mathbb{Q}(\sqrt{-631})$

-647:

x^23 - 20*x^22 + 3*x^21 + 37*x^20 + 46*x^19 - 2*x^18 - 151*x^17 - 384*x^16 - 610*x^15 - 849*x^14 - 995*x^13 - 955*x^12 - 798*x^11 - 596*x^10 - 378*x^9 - 202*x^8 - 75*x^7 + 21*x^5 + 19*x^4 + 13*x^3 + 3*x^2 - 2*x - 1

comment: $h(D)=23$; fundamental discriminant of $\mathbb{Q}(\sqrt{-647})$

-655:

x^12 - 18*x^11 - 41*x^10 - 47*x^9 - 55*x^8 - 43*x^7 - 32*x^6 - 19*x^5 - 15*x^4 - 4*x^3 - 6*x^2 + x - 1

comment: $h(D)=12$; fundamental discriminant of $\mathbb{Q}(\sqrt{-655})$

-671:

x^30 - 18*x^29 - 58*x^28 - 101*x^27 - 12*x^26 - 37*x^25 + 80*x^24 - 151*x^23 + 68*x^22 + 107*x^21 + 172*x^20 - 105*x^19 - 103*x^18 + 39*x^17 - 130*x^16 - 12*x^15 - 7*x^14 + 123*x^13 - 9*x^12 + 30*x^11 + 4*x^10 - 31*x^9 - 33*x^8 - 6*x^7 + 13*x^6 + 6*x^5 + x^4 + 7*x^3 - 4*x^2 - 1

comment: $h(D)=30$; fundamental discriminant of $\mathbb{Q}(\sqrt{-671})$

-679:

x^18 - 24*x^17 + 58*x^16 - 60*x^15 + 24*x^14 - 27*x^13 + 33*x^12 - 9*x^11 - 8*x^10 - 3*x^9 + 13*x^8 + 6*x^7 + 12*x^6 - 9*x^5 + 12*x^4 + 7*x^2 + 1

comment: $h(D)=18$; fundamental discriminant of $\mathbb{Q}(\sqrt{-679})$

-695:

x^24 - 16*x^23 - 118*x^22 - 443*x^21 - 1125*x^20 - 1926*x^19 - 2175*x^18 - 1631*x^17 - 1013*x^16 - 940*x^15 - 978*x^14 - 519*x^13 + 108*x^12 + 267*x^11 + 96*x^10 + 96*x^9 + 231*x^8 + 190*x^7 + 18*x^6 - 59*x^5 - 29*x^4 - x^3 - x^2 - 3*x - 1

comment: $h(D)=24$; fundamental discriminant of $\mathbb{Q}(\sqrt{-695})$

-703:

x^14 - 22*x^13 - 15*x^12 - 41*x^11 - 11*x^10 - 6*x^9 + 7*x^8 + 15*x^7 + 2*x^6 + 4*x^5 - 9*x^4 - x^3 - x^2 - 3*x - 1

comment: $h(D)=14$; fundamental discriminant of $\mathbb{Q}(\sqrt{-703})$

-719:

x^31 - 26*x^30 + 53*x^29 + 68*x^28 - 168*x^27 - 176*x^26 + 161*x^25 + 172*x^24 - 175*x^23 - 440*x^22 - 341*x^21 - 169*x^20 - 293*x^19 - 255*x^18 - 57*x^17 - 9*x^16 - 114*x^15 - 157*x^14 - 21*x^13 + 68*x^12 + 59*x^11 + 20*x^10 + 37*x^9 + 29*x^8 + 36*x^7 + 24*x^6 + 20*x^5 - x^4 - 11*x^3 - 11*x^2 - 4*x - 1

comment: $h(D)=31$; fundamental discriminant of $\mathbb{Q}(\sqrt{-719})$

-727:

x^13 - 25*x^12 + 23*x^11 - 41*x^10 + 15*x^9 - 30*x^8 - 8*x^7 - 4*x^6 - 6*x^4 + 5*x^3 - 8*x^2 + 4*x - 1

comment: $h(D)=13$; fundamental discriminant of $\mathbb{Q}(\sqrt{-727})$

-743:

x^21 - 26*x^20 + 22*x^19 + 33*x^18 + 86*x^17 - 515*x^16 + 895*x^15 - 825*x^14 + 509*x^13 - 451*x^12 + 701*x^11 - 897*x^10 + 832*x^9 - 642*x^8 + 491*x^7 - 387*x^6 + 274*x^5 - 156*x^4 + 68*x^3 - 23*x^2 + 6*x - 1

comment: $h(D)=21$; fundamental discriminant of $\mathbb{Q}(\sqrt{-743})$

-751:

x^15 - 25*x^14 - 12*x^13 - 53*x^12 + 9*x^11 - 21*x^10 + 4*x^9 - 2*x^8 + 29*x^7 - 17*x^6 + 12*x^5 - 7*x^4 + 6*x^3 - 9*x^2 + 5*x - 1

comment: $h(D)=15$; fundamental discriminant of $\mathbb{Q}(\sqrt{-751})$

-767:

x^22 - 24*x^21 - 60*x^20 - 187*x^19 - 260*x^18 - 74*x^17 - 20*x^16 - 136*x^15 - 152*x^14 - 174*x^13 - 74*x^12 + 111*x^11 + 34*x^10 - 156*x^9 - 131*x^8 + 34*x^7 + 81*x^6 - 3*x^5 - 39*x^4 - 5*x^3 + 9*x^2 + x - 1

comment: $h(D)=22$; fundamental discriminant of $\mathbb{Q}(\sqrt{-767})$

-775:

x^12 - 26*x^11 - 29*x^10 + 18*x^9 + 43*x^8 + 7*x^7 - 10*x^6 - 23*x^5 + x^4 + 9*x^3 + 4*x^2 - 5*x + 1

comment: $h(D)=12$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-31})$

-791:

x^32 - 30*x^31 + 55*x^30 - 34*x^29 + 171*x^28 - 186*x^27 - 278*x^26 - 726*x^25 - 1095*x^24 - 1098*x^23 - 1235*x^22 - 704*x^21 - 203*x^20 + 722*x^19 + 1729*x^18 + 2538*x^17 + 3118*x^16 + 3207*x^15 + 3052*x^14 + 2752*x^13 + 2381*x^12 + 1995*x^11 + 1558*x^10 + 1121*x^9 + 760*x^8 + 489*x^7 + 327*x^6 + 212*x^5 + 132*x^4 + 67*x^3 + 27*x^2 + 7*x + 1

comment: $h(D)=32$; fundamental discriminant of $\mathbb{Q}(\sqrt{-791})$

-799:

x^16 - 30*x^15 + 39*x^14 + 30*x^13 - 75*x^12 - 18*x^11 + 18*x^10 + 24*x^9 + 52*x^8 - 6*x^7 - 39*x^6 - 21*x^5 - 6*x^4 + 9*x^3 + 9*x^2 + 3*x + 1

comment: $h(D)=16$; fundamental discriminant of $\mathbb{Q}(\sqrt{-799})$

-815:

x^30 - 22*x^29 - 198*x^28 - 813*x^27 - 2024*x^26 - 3585*x^25 - 4669*x^24 - 4361*x^23 - 2694*x^22 - 466*x^21 + 936*x^20 + 1094*x^19 + 308*x^18 - 534*x^17 - 966*x^16 - 1075*x^15 - 1137*x^14 - 1365*x^13 - 1528*x^12 - 1444*x^11 - 1075*x^10 - 652*x^9 - 373*x^8 - 270*x^7 - 258*x^6 - 226*x^5 - 159*x^4 - 83*x^3 - 32*x^2 - 8*x - 1

comment: $h(D)=30$; fundamental discriminant of $\mathbb{Q}(\sqrt{-815})$

-823:

x^9 - 29*x^8 - 36*x^7 - 30*x^6 - 33*x^5 - 30*x^4 - 12*x^3 - x - 1

comment: $h(D)=9$; fundamental discriminant of $\mathbb{Q}(\sqrt{-823})$

-839:

x^33 - 34*x^32 + 93*x^31 - 329*x^30 + 796*x^29 - 1415*x^28 + 1618*x^27 - 1908*x^26 + 1977*x^25 - 1958*x^24 + 1127*x^23 - 979*x^22 + 372*x^21 - 71*x^20 - 497*x^19 - 26*x^18 - 530*x^17 + 108*x^16 - 155*x^15 - 188*x^14 - 204*x^13 - 267*x^12 - 3*x^11 - 104*x^10 - 26*x^9 - 126*x^8 - 52*x^7 - 41*x^6 - 9*x^5 - 12*x^4 - 15*x^3 - 10*x^2 - 5*x - 1

comment: $h(D)=33$; fundamental discriminant of $\mathbb{Q}(\sqrt{-839})$

-847:

x^10 - 34*x^9 + 67*x^8 - 12*x^7 - 43*x^6 + 10*x^5 + 23*x^4 - 12*x^3 + x^2 - x + 1

comment: $h(D)=10$; order of conductor $11$ in $\mathbb{Q}(\sqrt{-7})$

-863:

x^21 - 42*x^20 + 332*x^19 - 1316*x^18 + 3387*x^17 - 6531*x^16 + 10045*x^15 - 12933*x^14 + 14339*x^13 - 14012*x^12 + 12265*x^11 - 9789*x^10 + 7233*x^9 - 5008*x^8 + 3250*x^7 - 1946*x^6 + 1038*x^5 - 470*x^4 + 171*x^3 - 47*x^2 + 9*x - 1

comment: $h(D)=21$; fundamental discriminant of $\mathbb{Q}(\sqrt{-863})$

-871:

x^22 - 32*x^21 - 51*x^20 - 173*x^19 - 129*x^18 - 206*x^17 - 22*x^16 - 33*x^15 + 37*x^14 + 31*x^13 - 30*x^12 + 22*x^11 - 69*x^10 - 8*x^9 - 40*x^8 + 6*x^7 - 2*x^6 + x^5 - 9*x^4 - 11*x^3 - 9*x^2 - 2*x - 1

comment: $h(D)=22$; fundamental discriminant of $\mathbb{Q}(\sqrt{-871})$

-887:

x^29 - 32*x^28 - 98*x^27 - 95*x^26 + 265*x^25 + 6*x^24 - 13*x^23 - 247*x^22 + 268*x^21 - 349*x^20 + 413*x^19 - 405*x^18 + 574*x^17 - 839*x^16 + 866*x^15 - 953*x^14 + 951*x^13 - 820*x^12 + 758*x^11 - 628*x^10 + 447*x^9 - 336*x^8 + 217*x^7 - 115*x^6 + 70*x^5 - 29*x^4 + 7*x^3 - 7*x^2 - x - 1

comment: $h(D)=29$; fundamental discriminant of $\mathbb{Q}(\sqrt{-887})$

-895:

x^16 - 36*x^15 + 17*x^14 + 31*x^13 - 95*x^12 + 24*x^11 - 8*x^10 + 29*x^9 - 42*x^8 + 3*x^7 - 17*x^6 + 26*x^5 - 7*x^4 - 9*x^3 - x^2 + 4*x - 1

comment: $h(D)=16$; fundamental discriminant of $\mathbb{Q}(\sqrt{-895})$

-911:

x^31 - 38*x^30 + 54*x^29 - 324*x^28 + 682*x^27 - 1009*x^26 + 1574*x^25 - 2183*x^24 + 2267*x^23 - 2041*x^22 + 2736*x^21 - 4322*x^20 + 4854*x^19 - 3806*x^18 + 2937*x^17 - 3408*x^16 + 3938*x^15 - 3132*x^14 + 1783*x^13 - 1333*x^12 + 1700*x^11 - 1778*x^10 + 1182*x^9 - 524*x^8 + 300*x^7 - 371*x^6 + 397*x^5 - 283*x^4 + 136*x^3 - 44*x^2 + 9*x - 1

comment: $h(D)=31$; fundamental discriminant of $\mathbb{Q}(\sqrt{-911})$

-919:

x^19 - 37*x^18 - 17*x^17 + 77*x^16 + 15*x^15 - 183*x^14 - 147*x^13 + 63*x^12 + 112*x^11 + 29*x^10 + 4*x^9 + 23*x^8 - 6*x^7 - 51*x^6 - 42*x^5 - 9*x^4 - 5*x^3 - 10*x^2 - 5*x - 1

comment: $h(D)=19$; fundamental discriminant of $\mathbb{Q}(\sqrt{-919})$

-935:

x^28 - 44*x^27 + 212*x^26 - 270*x^25 - 298*x^24 + 952*x^23 - 1222*x^22 + 772*x^21 + 517*x^20 - 1837*x^19 + 2371*x^18 - 1853*x^17 + 558*x^16 + 896*x^15 - 1834*x^14 + 2075*x^13 - 1702*x^12 + 884*x^11 - 38*x^10 - 552*x^9 + 787*x^8 - 707*x^7 + 508*x^6 - 307*x^5 + 155*x^4 - 67*x^3 + 22*x^2 - 5*x + 1

comment: $h(D)=28$; fundamental discriminant of $\mathbb{Q}(\sqrt{-935})$

-943:

x^16 - 38*x^15 - 55*x^14 + 32*x^13 + 66*x^12 - 46*x^11 - 59*x^10 + 28*x^9 + 58*x^8 + 2*x^7 - 38*x^6 - 11*x^5 + x^3 + 8*x^2 + 5*x + 1

comment: $h(D)=16$; fundamental discriminant of $\mathbb{Q}(\sqrt{-943})$

-959:

x^36 - 40*x^35 - 35*x^34 + 194*x^33 + 515*x^32 - 1377*x^31 - 740*x^30 + 2568*x^29 + 1518*x^28 - 2434*x^27 - 3070*x^26 + 488*x^25 + 4158*x^24 + 1024*x^23 - 2989*x^22 - 970*x^21 + 778*x^20 + 726*x^19 + 198*x^18 - 954*x^17 - 35*x^16 + 911*x^15 - 77*x^14 - 320*x^13 - 144*x^12 - 101*x^11 + 217*x^10 + 93*x^9 - 75*x^8 - 3*x^7 - 11*x^6 - 3*x^5 + 7*x^4 - 7*x^3 + x^2 + 4*x + 1

comment: $h(D)=36$; fundamental discriminant of $\mathbb{Q}(\sqrt{-959})$

-967:

x^11 - 43*x^10 + 68*x^9 - 122*x^8 + 99*x^7 - 81*x^6 + 18*x^5 - 6*x^4 - 13*x^3 + x^2 - 2*x - 1

comment: $h(D)=11$; fundamental discriminant of $\mathbb{Q}(\sqrt{-967})$

-983:

x^27 - 44*x^26 + 60*x^25 - 463*x^24 + 860*x^23 - 1678*x^22 + 2483*x^21 - 2840*x^20 + 2620*x^19 - 1761*x^18 + 700*x^17 - 122*x^16 + 256*x^15 - 964*x^14 + 1531*x^13 - 1556*x^12 + 1071*x^11 - 474*x^10 + 121*x^9 - 118*x^8 + 274*x^7 - 380*x^6 + 347*x^5 - 234*x^4 + 116*x^3 - 41*x^2 + 9*x - 1

comment: $h(D)=27$; fundamental discriminant of $\mathbb{Q}(\sqrt{-983})$

-991:

x^17 - 43*x^16 - 23*x^15 - 64*x^14 + 36*x^13 + 7*x^12 - 121*x^11 - 54*x^10 + 50*x^9 - 14*x^8 - 62*x^7 - 21*x^6 + 14*x^5 - 11*x^4 - 18*x^3 - 4*x^2 - x - 1

comment: $h(D)=17$; fundamental discriminant of $\mathbb{Q}(\sqrt{-991})$

-1007:

x^30 - 50*x^29 + 228*x^28 - 171*x^27 - 739*x^26 + 1063*x^25 + 642*x^24 - 2904*x^23 + 468*x^22 + 3816*x^21 - 2916*x^20 - 2965*x^19 + 4638*x^18 + 738*x^17 - 3948*x^16 + 1488*x^15 + 2069*x^14 - 1844*x^13 - 303*x^12 + 972*x^11 - 465*x^10 - 198*x^9 + 349*x^8 - 122*x^7 - 111*x^6 + 110*x^5 + 13*x^4 - 34*x^3 + 3*x^2 + 4*x - 1

comment: $h(D)=30$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1007})$

-1015:

x^16 - 44*x^15 - 81*x^14 - 16*x^13 + 66*x^12 + 172*x^11 + 126*x^10 + 65*x^9 - 50*x^8 - 97*x^7 - 69*x^6 - 32*x^5 + 12*x^4 + 11*x^3 + 12*x^2 + 4*x + 1

comment: $h(D)=16$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1015})$

-1031:

x^35 - 46*x^34 - 60*x^33 - 79*x^32 + 427*x^31 - 949*x^30 + 2080*x^29 - 4134*x^28 + 6081*x^27 - 6076*x^26 + 4296*x^25 - 2342*x^24 + 1071*x^23 - 509*x^22 + 832*x^21 - 1863*x^20 + 1798*x^19 + 317*x^18 - 2117*x^17 + 1052*x^16 + 1732*x^15 - 2885*x^14 + 1284*x^13 + 891*x^12 - 1317*x^11 + 133*x^10 + 944*x^9 - 865*x^8 + 101*x^7 + 350*x^6 - 233*x^5 - 15*x^4 + 88*x^3 - 47*x^2 + 11*x - 1

comment: $h(D)=35$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1031})$

-1039:

x^23 - 49*x^22 + 45*x^21 - 28*x^20 + 4*x^19 - 76*x^18 - 228*x^17 - 40*x^16 + 34*x^15 + 17*x^14 + 24*x^13 - 19*x^12 + 19*x^11 + 62*x^10 + 48*x^9 - 31*x^8 - 26*x^7 + 14*x^6 - 6*x^5 - 25*x^4 - 14*x^3 - 4*x^2 - 1

comment: $h(D)=23$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1039})$

-1055:

x^36 - 44*x^35 - 258*x^34 - 1075*x^33 - 3055*x^32 - 5624*x^31 - 7937*x^30 - 9923*x^29 - 10740*x^28 - 10326*x^27 - 10987*x^26 - 12930*x^25 - 15727*x^24 - 19268*x^23 - 21853*x^22 - 21786*x^21 - 18965*x^20 - 13816*x^19 - 7586*x^18 - 1621*x^17 + 2783*x^16 + 5133*x^15 + 5664*x^14 + 4813*x^13 + 3207*x^12 + 1596*x^11 + 298*x^10 - 488*x^9 - 783*x^8 - 756*x^7 - 557*x^6 - 338*x^5 - 176*x^4 - 70*x^3 - 25*x^2 - 5*x - 1

comment: $h(D)=36$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1055})$

-1063:

x^19 - 51*x^18 + 27*x^17 + 7*x^16 - 88*x^15 - 15*x^14 + 63*x^13 - 55*x^12 + 9*x^11 + 60*x^10 - 36*x^9 - 42*x^8 + 7*x^7 - 39*x^6 - 45*x^5 - 8*x^4 - 10*x^3 - 9*x^2 - 1

comment: $h(D)=19$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1063})$

-1079:

x^34 - 62*x^33 + 558*x^32 - 2311*x^31 + 5554*x^30 - 8827*x^29 + 9963*x^28 - 8357*x^27 + 4465*x^26 + 47*x^25 - 3906*x^24 + 5450*x^23 - 4672*x^22 + 3217*x^21 - 2784*x^20 + 2770*x^19 - 3078*x^18 + 3181*x^17 - 2630*x^16 + 1485*x^15 - 620*x^14 + 207*x^13 - 283*x^12 + 465*x^11 - 501*x^10 + 274*x^9 - 17*x^8 - 152*x^7 + 182*x^6 - 118*x^5 + 28*x^4 + 17*x^3 - 18*x^2 + 6*x - 1

comment: $h(D)=34$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1079})$

-1087:

x^9 - 51*x^8 - 100*x^7 - 139*x^6 - 124*x^5 - 100*x^4 - 61*x^3 - 28*x^2 - 8*x - 1

comment: $h(D)=9$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1087})$

-1103:

x^23 - 50*x^22 - 244*x^21 - 524*x^20 - 89*x^19 - 129*x^18 + 28*x^17 - 515*x^16 - 165*x^15 - 274*x^14 + 16*x^13 + 106*x^12 - 257*x^11 + 19*x^10 - 140*x^9 + 19*x^8 + 34*x^7 - 53*x^6 + 14*x^5 - 33*x^4 + 17*x^3 - 5*x^2 - 1

comment: $h(D)=23$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1103})$

-1111:

x^22 - 54*x^21 - 81*x^20 - 149*x^19 - 107*x^18 - 62*x^17 - 217*x^16 - 252*x^15 + 85*x^14 - 237*x^13 - 192*x^12 + 40*x^11 - 128*x^10 - 68*x^9 - 70*x^8 + 45*x^7 - 95*x^6 + 21*x^5 - 15*x^4 + x^3 - 8*x^2 + 4*x - 1

comment: $h(D)=22$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1111})$

-1127:

x^24 - 56*x^23 - 64*x^22 - 512*x^21 + 14*x^20 + 448*x^19 + 896*x^18 - 217*x^16 - 1064*x^15 + 35*x^13 + 448*x^12 + 56*x^11 - x^10 - 126*x^9 + 71*x^8 + 8*x^7 + 21*x^6 - 28*x^5 - 21*x^4 - x^3 + 7*x^2 + x + 1

comment: $h(D)=24$; order of conductor $7$ in $\mathbb{Q}(\sqrt{-23})$

-1135:

x^18 - 54*x^17 - 235*x^16 - 439*x^15 - 512*x^14 - 354*x^13 - 64*x^12 + 199*x^11 + 370*x^10 + 389*x^9 + 228*x^8 - x^7 - 144*x^6 - 131*x^5 - 43*x^4 + 9*x^3 + 8*x^2 - x - 1

comment: $h(D)=18$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1135})$

-1151:

x^41 - 64*x^40 + 242*x^39 - 128*x^38 + 67*x^37 - 1356*x^36 - 1996*x^35 + 1602*x^34 + 1673*x^33 + 4366*x^32 - 273*x^31 - 6530*x^30 - 197*x^29 + 10*x^28 + 1681*x^27 + 716*x^26 - 2057*x^25 + 885*x^24 + 2067*x^23 + 291*x^22 - 1309*x^21 - 210*x^20 - 327*x^19 + 197*x^18 + 144*x^17 - 100*x^16 - 33*x^15 + 207*x^14 + 33*x^13 - 229*x^12 + 128*x^11 - 26*x^10 + 49*x^9 + 32*x^8 - 50*x^7 - 59*x^6 + 65*x^5 + 3*x^4 - 20*x^3 - 2*x^2 + 5*x - 1

comment: $h(D)=41$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1151})$

-1159:

x^16 - 64*x^15 + 190*x^14 - 191*x^13 - 106*x^12 + 245*x^11 - 95*x^10 - 137*x^9 + 114*x^8 + 34*x^7 - 106*x^6 + 32*x^5 + 22*x^4 - 23*x^3 - x^2 + 5*x - 1

comment: $h(D)=16$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1159})$

-1175:

x^30 - 60*x^29 - 176*x^28 - 128*x^27 + 768*x^26 - 1024*x^25 + 130*x^24 + 780*x^23 + 80*x^22 - x^20 - 185*x^19 - 789*x^18 - 612*x^17 + 32*x^16 + 66*x^15 + 160*x^14 + 333*x^13 + 359*x^12 - 24*x^11 + 30*x^10 - 80*x^9 - 23*x^8 - 54*x^7 - 21*x^6 + 9*x^5 + 15*x^4 - 11*x^3 + 2*x^2 + 3*x + 1

comment: $h(D)=30$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-47})$

-1183:

x^14 - 64*x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + x + 1

comment: $h(D)=14$; order of conductor $13$ in $\mathbb{Q}(\sqrt{-7})$

-1199:

x^38 - 70*x^37 + 298*x^36 - 1333*x^35 + 4478*x^34 - 10837*x^33 + 18462*x^32 - 24217*x^31 + 22355*x^30 - 10334*x^29 - 8896*x^28 + 27317*x^27 - 38888*x^26 + 40346*x^25 - 32907*x^24 + 20669*x^23 - 6386*x^22 - 7422*x^21 + 16831*x^20 - 19069*x^19 + 14744*x^18 - 6914*x^17 - 153*x^16 + 3408*x^15 - 3285*x^14 + 1854*x^13 - 601*x^12 + 142*x^11 - 321*x^10 + 548*x^9 - 563*x^8 + 445*x^7 - 262*x^6 + 98*x^5 - 17*x^4 - 2*x^3 + 3*x^2 - 1

comment: $h(D)=38$; fundamental discriminant of $\mathbb{Q}(\sqrt{-1199})$

Definition

For $D<0$, $D\equiv1\pmod 8$ and $3\nmid D$, put $m=-D$. Let $\omega_D=\mathfrak f(i\sqrt m)/\sqrt2$, with $\mathfrak f$ Weber's function [2] [3]. $W_D(x)\in\mathbb Z[x]$ is the minimal polynomial of $\omega_D$.

Parameters

$D$

— discriminant ($D<0$, $D\equiv1\pmod 8$, and $3\nmid D$)

Formulas

(1)

$\mathfrak f(\tau)=\eta(\tau)^2/(\eta(\tau/2)\eta(2\tau))$, where $\eta$ is Dedekind's eta function [4].

(2)

For $m=-D$, $j(i\sqrt m)=((256\omega_D^{24}-1)^3)/\omega_D^{24}$ [6].

Comments

(3)

The order $\mathbb Z[i\sqrt m]$ has discriminant $4D$, while the polynomial is indexed by the discriminant $D$ of the order whose ring class field is generated by the Weber class invariant [1]. This is why the parameter is $D$, not $4D$.

(4)

Each entry's comment gives the class number $h(D)$, which is the degree of $W_D$, and says whether $D$ is fundamental.

(5)

These are smaller class polynomials than the Hilbert class polynomials for the same discriminants. They are used in the CM method because Weber class invariants have much smaller height than the corresponding singular moduli.

(6)

For the entries here, PARI's polclass(D,1) [5] is the minimal polynomial of $1/\omega_D$ or $-1/\omega_D$; the sign depends on $D$. For example, polclass(-23,1) returns $x^3-x^2+1$, which corresponds here to $W_{-23}(x)=x^3-x-1$.

Programs

(P1)

PARI/GP

\p 100
D = -23;
P = polclass(D, 1);
w = weber(I*sqrt(-D))/sqrt(2);
Q = polrecip(P);
S = polrecip(subst(P, x, -x));
if (abs(subst(S, x, w)) < abs(subst(Q, x, w)), Q = S);
Q / pollead(Q)

(P2)

Sage

from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.real_mpfr import RealField
from sage.rings.complex_mpfr import ComplexField
from sage.libs.pari import pari
from sage.schemes.elliptic_curves.cm import hilbert_class_polynomial

D = -23
R = PolynomialRing(ZZ, ('x', 'j'))
x, j = R.gens()
H = hilbert_class_polynomial(4*D).change_ring(ZZ)
relation = (256*x**24 - 1)**3 - j*x**24
factors = H(j).resultant(relation, j).factor()
w = ComplexField(200)(pari('weber(I*sqrt(%d))' % -D)) / RealField(200)(2).sqrt()
min((p.univariate_polynomial() for p, e in factors), key=lambda p: abs(p(w)))

References

[1]

David A. Cox, Primes of the form $x^2+ny^2$: Fermat, class field theory, and complex multiplication, 2nd edition, Wiley, 2013.

[2]

Reinhard Schertz, Weber's class invariants revisited, Journal de theorie des nombres de Bordeaux 14 (2002), no. 1, 325-343. (doi)

[3]

Noriko Yui and Don Zagier, On the singular values of Weber modular functions, Mathematics of Computation 66 (1997), no. 220, 1645-1662. (doi) (MR)

Links

[4]

DLMF 23.15: Dedekind's eta function

[5]

PARI/GP manual: polclass

[6]

Wikipedia: Weber modular function

Similar tables

Hilbert class polynomials $H_\Delta$ — store the class polynomials of the $j$-invariant; the entries here are smaller class polynomials for Weber's function

Ramanujan's class invariants $G_n$, with Weber's $\mathfrak f(\sqrt{-n})$ — stores $G_n$ and $\mathfrak f(i\sqrt n)$; for $n=-D$, the roots of $W_D$ include $\omega_D=2^{-1/4}G_n$

Modular polynomials for Weber's $f$-function — stores modular equations for the same Weber function

Rational singular moduli — Formula (2) relates the Weber class invariant to the singular modulus $j(i\sqrt m)$

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every negative discriminant $D\equiv1\pmod8$ with $3\nmid D$ and $|D|<1200$)

How they were obtained:

For each $D$, the generator computes the exact Hilbert class polynomial $H_{4D}$ and the exact resultant of $H_{4D}(j)$ with $(256x^{24}-1)^3-jx^{24}$. It factors this resultant over $\mathbb Z$, computes $\omega_D$ from the eta quotient in Sage's arb-backed ComplexBallField, and selects the unique factor whose interval evaluation at $\omega_D$ contains zero.

The selected factor is monic over $\mathbb Z$. Each polynomial is then compared with PARI's polclass(D,1) [5] after the reciprocal transform and the sign change in $x$ when PARI's chosen Weber class invariant has the opposite sign.