curve #4225

y² + xy + y = x³ − x² − 1207026666637504238666072016967080x + 16771556035482880308475507184754550232016679133547

a-invariants: [1, -1, 1, -1207026666637504238666072016967080, 16771556035482880308475507184754550232016679133547]
c-invariants: [57937279998600203455971456814419825, -14490624414657208325805078213927015848590855106495625]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 432634656575362589579070369184686409693027916184207096152588384540341013487139150
discriminant (Δ): -8969028904260421044394593515425750923399144281146323244274051703131292140928254003846697826304000000
Faltings height: 17.9427
naive height: 240.2107
primes of bad reduction: 2, 5, 7, 13, 23, 29, 151, 23509, 175294403, 587530212168689, 389919287788835426706086699226136196124383663
regulator: 65499564227554616205782.2181839134614846617048236024388628859986680173578326625109246
submitted by: apkorte
submitted at: 2026-10-14 13:47:08 UTC
last updated: 2026-10-14 13:47:08 UTC

Witness: 23 independent points log in to add more points →

(17004825668196269, 1078639428254540867167865)
(17700365641176665, -975868524389178964106527)
(102472574450902349, -31130545554285624043451975)
(21824230955068669, 907713745868819464965865)
(44522189580945749, 7161361541751478417893025)
(87981914423836269, 24323382972850911770831865)
(19848813471369485, 795897613096317262279993)
(34197468676509013, 3935366021431245392796321)
(66203642599144289, -15067428979061276511505295)
(10355260681186454349, 33322617900744033461371229625)
(16472368076479099, 1165574058165964330096275)
(22182507497877539, 954925803746689922945755)
(20024886861791869, -794282710200824353378535)
(-37286019294254611, -3152775012253100739013895)
(50131629246674669, -9069242731171508992066535)
(9624390279919484141/289, -18057309283150806554963744855/4913)
(-3463652833636211, -4572824541623819819895495)
(1484089323894909409/81, -654888624363616554339304555/729)
(5861000163903669, -3146189071272752746929535)
(21336181796212249, -855068604925835612239775)
(4229397145339556899/121, -5533914724475629864114590135/1331)
(-253714308232866059/9, 143875182550265553120227075/27)
(30879840773990349, 2990766898340767132060025)

Commentary

Specialization at T = 243/47 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -48788204994959095721152T^8 + 292967918358775673142176T^7 - 257383034328976566251548T^6 - 697546673797396881668416T^5 + 120168749817393327515900T^4 - 1757567929634520292525024T^3 - 413537200758726229861444T^2 + 1707114327506779370979968T - 562162388405818919328652, A6 = 4093740998135849949800603694210304T^12 - 39254200079660475614530561864102528T^11 + 106574491886043297526892628627228560T^10 + 93605566178607331448553679365937280T^9 - 784525015208748993169852391016411996T^8 + 155163719535379155266150397777650688T^7 + 2578710394897183836702469318351911360T^6 - 11129400918516684422195231027746944T^5 - 2571149805127224053198426441759406936T^4 + 885351238544731577742776581975479680T^3 + 1010399979603017167929810575585795728T^2 - 806575307806361814743442724032893696T + 165741056816888627629082643287877460. It is a 2-neighbour of the fibration given in curve#4191 (itself a 2-neighbour of the fibration in curve#3900): its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-14 13:47:08 UTC · history