curve #3125

y² + xy + y = x³ + x² − 23671323727237963827709029432581002734x + 32479411170253864847996755216329958698104110437129185946

a-invariants: [1, 1, 1, -23671323727237963827709029432581002734, 32479411170253864847996755216329958698104110437129185946]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 5656989953868346894737119189064534826653330489
discriminant (Δ): 393160505905096073124817180853595652017675372111508294026768797023020288098616045745597219777293541382057983342929
Faltings height: 20.5050
naive height: 269.7856
primes of bad reduction: 3, 7, 11, 17, 41, 43, 59, 73, 89, 179, 181, 197, 293, 353, 433, 443, 1181, 2269, 2389, 2629511
regulator: 724633148635.254317359380573736557578609258503919043790985666
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:27 UTC
last updated: 2026-09-26 20:38:27 UTC

Witness: 11 independent points log in to add more points →

(3735369604870046796871/225, -219119146931097581659186027238264/3375)
(1419369180780813229169/1024, 50035539819520401352479158510457/32768)
(17377249838302980740598031/106276, 72407047518464979439091470104220720189/34645976)
(706125819281986395038789311/488012281, -12090441220491174083174609198006377369316/10780679299571)
(3154265868324388601501479/410881, 4585925489897778387670082268236452000/263374721)
(2611202057170942279981170576988/463983032569, -2782409965932542725463409330862656828121560265/316048074413797747)
(-25287172410358202760635/104976, -210127032384335358035509430843486417/34012224)
(1522931485637300523270329195719675261073886157331825570747896485415/114551486160991949376813174104106574629330572356, -1762935125675040714113465785413168208467097084999554610446355445159073910029287898115436454420028973/38770472334254900926053753710170117893821151854168194428335882650519496)
(2004257467874308134963809506820183110665756995994304986213312320841307687/55034045011542821423623857105136389356824319174168449, -2812990962686450731121016980197805840534882811886805421008906724111156921898304231917429288742872906067881808/12910621587072220361535141180555623704106882010802404046535277250371804355670207)
(36792105757070068154023089375849558292/2039496998193748921, -215538575605909644475417080891711964685007852415386475727/2912625219857208513938503469)
(861845476423155632349514348771472734858439789186135814491550892365/24088345470803328858530377469994206813417721616, -792953874783035143199987928715566898299924247951264101152586173750986994682184836062644285464915847/3738612525720859571942905656829310315764169694368145909539261668569664)

Commentary

2-isogenous image of D-z4old9101112-017, rational kernel x=2722011719392931145. Source: https://dujella.github.io/tors/z4old9101112.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 20:38:27 UTC · history