curve #1378

y² + xy = x³ − 1148701945424064481425859592208024187458738800x − 13425175949192314803770353966641412474743403547822354677669726560000

a-invariants: [1, 0, 0, -1148701945424064481425859592208024187458738800, -13425175949192314803770353966641412474743403547822354677669726560000]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 358210529915244759591648810
discriminant (Δ): 19145098994861964429545895795064363715838657765757988550449924211657419900402361867585237450291573423636384944073772803768309427776000000
Faltings height: 24.8824
naive height: 322.8785
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 139, 593, 859, 11503, 1563539
regulator: 2561826934.968044209846524272240591056019182854
submitted by: jesper-petersen
submitted at: 2026-09-24 19:00:19 UTC
last updated: 2026-09-24 19:00:19 UTC

Witness: 4 independent points log in to add more points →

(3459372024841640952411837154405815117432862646708103372784/22320631498402761986540384963272225, -198166875500239066433596644069566652861371257081638422683928762045035015959900308373872/3334722626977077116979387646264686004463160148776625)
(-115849515182759448989118228092918740276199661207808871474957819631768043559683817420464841010324095039078104454477848823277387848275856/7177727205270570057746402633715884567999175196751121949519151718858920049446788130313934713665091215319555172249, 580244574731810460967718608054886189082981303485571028617212680759227438612191302417641875585308158725101443369371483065893355232243736357819347836117048477008680905980799173081404770831785279927572144/608107588495791235839739509161637523196274644747714670952516785938680757464933092288979154458488981008095930013423934142416609367270860988093742568337928915425680490307)
(399593379739314574336578547019108773488466809009931469065232613219636499493968973439911749726509784763544841988240016243347647684217175179663219448821344/8056177872254582676980706294577184259997234695478874763716904136510115910356477716889815941369550824869725733776299860412067352025, -5195632564736435155903361660859355514240271363618206505869782031129352137039043003915589321502375308511303533638028863092326206650098994264543163941747663382581791732016221431078692267727738970496168148045154251813970567966626832/723092021562655744601947615885004547901859369927552335642016188868564374809885164677688818157230304157648672821651940428520588960478405831853311398574120882027536560006238230194911466697766283875)
(-41136470860494839377794709142319440747656047486842241379752276249994046871120406631653433087635938608941883821145614241276160384794406493856861738482833480029478648284716112606322194883881112672249500/2819113211622475213320884009337927744657006130928310486369538617448777835685135282393420895021956488506885600928880914483821638175419358861189230166553956997493804986052917113921, 71730912758312799723931275798837728830101076785547064756182350162149685348656285854428920497270283177306765831736570543133114700641239310895260347516362681020646491721152171496891669933677038486700897045569949901313887119089789437064670549380873363012430497599084339999480979215845900292452727367200/149681723548709951482699628507792218991645927590138611435744784788839148481660143172914988986614976027381840157934679904444575599240053904942476446632901698148864455483478245069016921675733240367982362214552494711705104603084059648991649094764812076430859227844144031)

Commentary

Obtained from ICARM curve #1105, a rank-at-least-4 curve with torsion \(\mathbf Z/12\mathbf Z\), found by Tom Fisher (2008). Quotienting by its unique rational point of order 2 gives a 2-isogenous curve whose global minimal model has ainvs \([1,0,0,-1148701945424064481425859592208024187458738800,-13425175949192314803770353966641412474743403547822354677669726560000]\), torsion \(\mathbf Z/2\mathbf Z\times\mathbf Z/6\mathbf Z\), and conductor \(358210529915244759591648810\). The four independent source points were mapped through the isogeny, yielding four independent points on the quotient. The isogeny, minimal model, conductor, torsion and image points were verified in Magma.

last edited by jesper-petersen at 2026-09-24 19:00:19 UTC · history