curve #4227

y² = x³ + x² − 68127108873608535661397469390258798996860x + 6532808944893560444441297397602161282271941646259375386480400

a-invariants: [0, 1, 0, -68127108873608535661397469390258798996860, 6532808944893560444441297397602161282271941646259375386480400]
c-invariants: [3270101225933209711747078530732422351849296, -5644346928388036224016901558883866606153440053552494868030161344]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 207479491880304957471326786939605942587517259346001379219559248730591334381853263329044682907789019286360
discriminant (Δ): 1799987296794708805147241090174322551863878104622532848801918616074317870058655341928888242110152543001229725760064116000000
Faltings height: 22.4083
naive height: 293.6802
primes of bad reduction: 2, 3, 5, 7, 19, 31, 43, 103, 20231, 23899, 3569885381, 9126168707, 1544303695164929, 3892264626982588260567838467320645992725385851005177
regulator: 141743139184419384275617828872.80119308433360541022928556721936010607230067793483675952805838
submitted by: apkorte
submitted at: 2026-10-14 17:52:07 UTC
last updated: 2026-10-14 17:52:07 UTC

Witness: 27 independent points log in to add more points →

(242387149450676369440, 2064046792180316916479018260140)
(214981763702437237975, -1350025636508247323583061245870)
(203350902983094556480, 1043064212814658617469432011900)
(5104108126103493955, 2487009004145180083387254870000)
(280021453218802005760, -3068029187961607987849487581980)
(-232166661576233530370, -3136168659383722097604615815550)
(121159210339899257890, 239051107760016009690251766810)
(798201115786005316330, -21464099736007339740188135274750)
(12535116125798032387180/169, -3018558484826354943547594398476220/2197)
(180338519522469952210, 334414427335281314129933093430)
(120020344916946664288, 291631426836744781701921888756)
(122137602093176362930, -184191125647151756975180212650)
(318034406781641660320, -4127216431458214423535506703340)
(9985924876717556898417580, -31556035917386929429800826970014618500)
(122832719276167294380, -133625807010851586517607864700)
(246310779922287622675, -2166978635529774426600988889820)
(-275229689796252339890, -2105797373298935596001947138830)
(7463723375425311616378330/66049, 8938029918597747932555077442761065330/16974593)
(-298190149193808271670, -577372796430234948807228798750)
(-83053025459624470550, -3408531630369254037192501910530)
(-97611108208497964052, -3500391390483684957435386506596)
(-281563015054303308032, 1842065697402283527899876994276)
(-108172787006488986828788/529, -41995361347862145501558120365597772/12167)
(371912723358334383820, -5712975912515585453996385224340)
(455389096735081309915870/2209, -116051278744402913250171342006106050/103823)
(32841105338499687091345/64, 5288540213301408773229821751275175/512)
(187367314434910605310, 588071517821208175332157606530)

Commentary

Specialization at T = 433/670 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -5359397556934326557952T^8 - 135465106160634648571008T^7 - 1009560857120332086707244T^6 + 302645697651446768292456T^5 + 267102270709279813507320T^4 - 18710244598364193145536T^3 + 1090445903163236801932416T^2 - 91760564814653904426432T - 567824854400092808778672, A6 = 156065240689863717949223596360704T^12 + 4827291258544591401284174444461824T^11 + 48077643204571887106158625640791056T^10 + 215456560411136671924188808659259680T^9 + 998209464630408701484689228105999700T^8 - 909644941490611250119006829109042720T^7 - 2569542697812503924009268807282804720T^6 + 1241192156761670225595583604913427680T^5 + 2544459298635871164419215332497359200T^4 - 725030770745235523201629773993683200T^3 - 1216067410812944483775200857080372480T^2 + 159116305290413916148713260094977280T + 238920668104630371120386669007676800. It is a 2-neighbour of the fibration given in curve#4205 (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 17:52:07 UTC · history