curve #3191

y² + xy = x³ − 2559847023411566099401004283575x + 2583072824458606980138635616347507315247155625

a-invariants: [1, 0, 0, -2559847023411566099401004283575, 2583072824458606980138635616347507315247155625]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 159345113070803533282965684718748208769606387290
discriminant (Δ): -1808869227102027526818691586465214543912783913685197936388516424895492022913699671703552000000
Faltings height: 16.5904
naive height: 222.6538
primes of bad reduction: 2, 3, 5, 7, 23, 61, 73, 79, 101, 149, 263, 421, 547, 701, 757, 2719, 3863, 406591
regulator: 5061724094928447.86536846476778683914809190455666274531200012614206
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:57 UTC
last updated: 2026-09-26 20:41:57 UTC

Witness: 14 independent points log in to add more points →

(2317987475995054950/2809, 4784053020734564720290045425/148877)
(-135359718875820650/169, -141011684907674424239578775/2197)
(990481658847361084150/96721, 30827484657361108654844033030675/30080231)
(1564003630309154522230/815409, 50637818982549047203807602996035/736314327)
(12521158632781366482550/9690769, 1141830772288763694365372086986125/30167363897)
(23134829071213419850/537289, -19584739648005483648083747184275/393832837)
(-1816988081087082791610/1261129, 81116176945161752083269522228855/1416247867)
(46946934498530237350/674041, 27139235822539997024780869989425/553387661)
(-72345105882333397532250/53304601, -23211849321560017917796276200734175/389176891901)
(224205620575457611653325963774870/278778321916912129, -4757158129935260714004968426763696651090404887285/147193454981092656875482367)
(410226844347958499517298342150/361790109849961, -232176526066635477043164430894976406189525575/6881530447422051039541)
(31650312470782155581959218538450/6876638052697009, -169452486715975319212977089925043397650679641175/570248628209011661738473)
(441293673628728942208090/156225001, -261182136159175522519925568980089025/1952656287499)
(-74277356605452482449548042542645/116409713995387684, -2498318450497054024789950446564633240809743828215/39717676424444422001441752)

Commentary

2-isogenous image of D-z2old141516171819-071, rational kernel x=15742887693283151/4. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:41:57 UTC · history