curve #312

y² + xy + y = x³ − x² − 6568019580778599489585141337989692763602574718655x + 6357048702469147640200079650140599226292653501480652916481851397620447847

a-invariants: [1, -1, 1, -6568019580778599489585141337989692763602574718655, 6357048702469147640200079650140599226292653501480652916481851397620447847]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 565363502863493379565569563649307108299188776386222378070363275106421667133549554450
naive height: 348.8325
Faltings height: 26.9627
discriminant (Δ): 675555591393920126210578335859174012733377876129576927329773074052725622143925022750267503150613919256894560960666950209537112482403220443264000000
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 47, 61, 71, 79, 83, 101, 127, 137, 229, 241, 293, 311, 523, 541, 827, 1009, 1753, 3331, 4091, 6637, 9007, 16069, 44263, 52609, 67157
regulator: 55334661388047328635264.233241137894046053441742156612656109565455169
submitted by: 7fff-zip
submitted at: 2026-08-24 00:23:57 UTC
last updated: 2026-08-24 00:23:57 UTC

Witness: 15 independent points

(46664353179227656746344209195170267/5869551769, -9609414434353093913425768111368704314373409144654338/449683969678397)
(20500362356964924142848252139122731/1440733849, 2890519800749380699697383530571870546009094699988270/54685934706493)
(427407679394957380896328721938730067509/240947043855961, -2006826744292694090548290082282236945466283191949070963840/3740093018895795087709)
(103167544390400490835793902896668795/62785826041, -554359388049341210240512232520526671728133838978014/15732307216919411)
(134729047327530661756060101/25, -1411330887440267498472117124542946898064/125)
(717049180243690501327959841006625/354907921, -7723018542827921882484225749198651619683288538396/6686110323719)
(3259714524420718295605994459353331/1552911649, -82539375370618395063776110058719841807163463349430/61195589352143)
(1693650329908056916337234543972315/87179569, -69121783083240021677521702496053712441134400754162/813995635753)
(536301036363160960997997589169635/75325041, -11703903112195500857742495776305395631020516683398/653746030839)
(6538582364882675844199175387060159/1771561, 528719457855062509908391094465081265327115697049090/2357947691)
(640676361896207555519981883276618853401/101299655541529, 15046710442381457969025156385343767062708474199448217868380/1019558038003681457917)
(19021380651749437931176718556811035/10293928681, -760212660155743957404916573496430843743000260111566/1044411710045579)
(231690727121059545502326343077/61009, -90646366578917143905881190906667010579665192/15069223)
(32672882199157031838907171/9, 148764467232755722042553420072889834730/27)
(4428458137325117467968668572701367/2582977329, -48397502483322969510165969342065192368038716287990/131274656791767)

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