curve #3422

y² = x³ + x² − 1403690649658805063822079426565364532330941688777465x + 20330956262404043956056670578860396389806578729906895240267998011765438043400

a-invariants

[0, 1, 0, -1403690649658805063822079426565364532330941688777465, 20330956262404043956056670578860396389806578729906895240267998011765438043400]

rank (lower bound)

≥ 30

torsion subgroup

trivial

conductor (N)

52134320873208521256274675048146563368430334659636792268999245525914507683535426389573186630918936299480067995762355412351638762223020

discriminant (Δ)

-1557712265487044513546167827143913520669054255412451105936788599224797766408660085753236928614411727616960886037349792101852492229889744836053923648750000

Faltings height

28.2415

naive height

364.9352

primes of bad reduction

2, 3, 5, 11, 17, 43, 53, 157, 5707205470237, 67715132647410929562018040303594374120854408406577007, 33602968456535698041735272839588813420960494298973891245583

regulator

4009772939735830775446536626582013866.7355636289950151681394726785061884130153234302916612632528844

submitted by

Jason Ma

submitted at

2026-09-27 00:03:08 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-27 01:31:45 UTC

Witness: 30 independent points log in to add more points →

(459620814787170159712743705/64, -52764913032914496864752477053386921287325/512)
(6597963351771371456807822061/529, 839758206891464477238524332634955717931305/12167)
(6955136331364073428906843201485/14884, -18285279746234302711976570356705168018264683165/1815848)
(11780804662995131736641355, 73684310892975053809039413780161744385)
(134887586139639859543961205/4, 852037554797171420283134248601718413775/8)
(-43190513477507919377580567, 19713165792651836233302314685107127591)
(23543284617599072974045095, 18253403506614031663591367634075119775)
(4837730743885402367709697125/361, -429814356194667702520051028155622175085605/6859)
(140870438889370879633833795, -1618052262257438010474730337347265111775)
(1498067865785906286203267145/64, -8857006901353011785592175632573018707355/512)
(4717487725470377418087698065/9, -323209957899763545554297486913350654436445/27)
(3231137724313981743040131460065/118336, -1977816053315020372814357742790297685243591055/40707584)
(11674521938077037702575736520/361, 641900174013235640628873061674929593524000/6859)
(1611384769868085244437015057789/62500, -558730587152811742984467205029229459310028837/15625000)
(4500121321571542075690872454845/209764, 916039211838665721840020743867436749335985755/96071912)
(-3507363114518669510144432625/169, 442267455300870056846725257541172426089035/2197)
(14091415382015090329662335739/625, -187506118506675832870700217837515059801737/15625)
(-24693843840639970261281686312619327/677508841, 2678706971386587357757138385112919139785650955384269/17634877622389)
(-7781415874779819401277532395/289, -965322041841784561344495874846564743831375/4913)
(133987763190167275592242561545/8281, -32455684808645981302195483213503463088077125/753571)
(-3383302461076307727167797320/841, 3926009990487321348381826703795405155720140/24389)
(20906561753347079006790195, 11069843318076081600669951532750233375)
(-169004028306952485493986795/4, -519323621423312328684216225939684360225/8)
(5592057056832806790317599390360095/1315440361, -5733222828553009453086291985535534255512488055212075/47709706453109)
(-805892388986176650963759023818635735/34488575521, -1286928147862193617021070901761115784205129760101930295/6404907848580431)
(123582807707654929127744515809/5776, 4220639379918000019411770953970703078312185/438976)
(-344437474531258027701458484557955/13293316, -9609041560899716585194511737920053641940497322325/48467430136)
(26722924706098550083445702402913615/1306750201, -629763828121383625106170106686641995171950426276845/47237713015949)
(-9301906701774077307539640, 180507905340281050982949129055958078040)
(473979832403772094776633405/49, 30016056403492813029056551889006756338425/343)

Commentary

Specialization of a rank-17 elliptic fibration on the K3 surface X1092 (lattice L_546). Selected by a full-box Mestre-Nagao sieve (primes <= 5000, |n|, d <= 10^5) with exact rescoring, then a 2-covering search seeded by the 17 generic sections. Under GRH+BSD the rank is exactly 30 (explicit-formula bound 31.93, global root number +1).

last edited by wgxli at 2026-09-27 02:02:47 UTC · history