Baran [10.1016/j.jnt.2014.05.017, MR:MR3253304] showed that this curve admits an exceptional isomorphism to the modular curve $X_{\mathrm{sp}}^+(13)$.
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ - x^{3} z + x^{2} y^{2} + x y^{3} + 2 x y^{2} z + 2 x y z^{2} + 2 x z^{3} + y^{3} z - y z^{3} $ |
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This modular curve has 7 rational CM points but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve |
CM |
$j$-invariant |
$j$-height | Canonical model |
| 49.a2 |
$-7$ | $-3375$ |
$= -1 \cdot 3^{3} \cdot 5^{3}$ | $8.124$ | $(0:-1:1)$ |
| 256.a1 |
$-8$ | $8000$ |
$= 2^{6} \cdot 5^{3}$ | $8.987$ | $(0:1:0)$ |
| 121.b1 |
$-11$ | $-32768$ |
$= -1 \cdot 2^{15}$ | $10.397$ | $(-1:1:0)$ |
| 361.a1 |
$-19$ | $-884736$ |
$= -1 \cdot 2^{15} \cdot 3^{3}$ | $13.693$ | $(1:0:0)$ |
| 49.a1 |
$-28$ | $16581375$ |
$= 3^{3} \cdot 5^{3} \cdot 17^{3}$ | $16.624$ | $(0:1:1)$ |
| 4489.b1 |
$-67$ | $-147197952000$ |
$= -1 \cdot 2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3}$ | $25.715$ | $(0:0:1)$ |
| 26569.a1 |
$-163$ | $-262537412640768000$ |
$= -1 \cdot 2^{18} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3} \cdot 29^{3}$ | $40.109$ | $(3/2:-3/2:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 78 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{-2323513461112531990107446305456128x^{21}-18335113415020737557188704299999232x^{20}y-496692006692764985852126533884254336xy^{20}-89629989405629157676149588048000y^{21}-78219520097628650234172847670568960x^{20}z-171534015347790011179139151876573696x^{19}yz-4422880159390699840224932940779042816xy^{19}z-494587377427132370521263129595828736y^{20}z-333561447328458914617911407633092608x^{19}z^{2}-238088229569145215777504282023885824x^{18}yz^{2}-19012141396937334722704086755250616736xy^{18}z^{2}-3909272467959980046040697085159349920y^{19}z^{2}-177282310187878206265916875935263296x^{18}z^{3}+300518605601742364252308234107555584x^{17}yz^{3}-58344561867542959715313391724948978496xy^{17}z^{3}-13008058283087489015361106243774316160y^{18}z^{3}+141278691043899920087723759670439520x^{17}z^{4}+187123989415301499255671532187729256x^{16}yz^{4}-147389406607187113611691430755570181296xy^{16}z^{4}-26466947600520277427509651626779871864y^{17}z^{4}-831395903945870067718445916516755048x^{16}z^{5}+65788585084810084840119884620712112x^{15}yz^{5}-162632634978120226778520296965838142528xy^{15}z^{5}-43119619410841126004896611450805787080y^{16}z^{5}-217876745079722850146802407513539939x^{15}z^{6}+114062179328628686191275153284815541x^{14}yz^{6}-128294295293064688078555935374634377294xy^{14}z^{6}+86352296973421633297363771384163012058y^{15}z^{6}+1164561814695480028021514140847115626x^{14}z^{7}-7270249650597624884980120605254550395x^{13}yz^{7}+877083782484484854912398720273695661389xy^{13}z^{7}+217893659458898314669859783204279607768y^{14}z^{7}+13904546582333995137230364504136282924x^{13}z^{8}-22800206580226113613378844922539095783x^{12}yz^{8}+2610193542530409902086106287410829149642xy^{12}z^{8}+920571651797972700830637192076768333051y^{13}z^{8}+22597626053175858576735824705635768724x^{12}z^{9}-63315606008644606876338162020485529849x^{11}yz^{9}+5579522700604964541311908441761693982830xy^{11}z^{9}+1642103081445517980612708854284185140132y^{12}z^{9}+27334478482785271449286151498381857380x^{11}z^{10}-85918386210285802945171169753023455428x^{10}yz^{10}+9624217414870710278029942478135947377680xy^{10}z^{10}+2583203869896836576137352412712253011279y^{11}z^{10}-15138029107667000561149307531469410547x^{10}z^{11}+4856199670719514527126598331400883979x^{9}yz^{11}+7810048535587506078229043940287172011674xy^{9}z^{11}+6049181325751677406252242381779299205183y^{10}z^{11}-154699844346706950361305550052881626866x^{9}z^{12}+149365168292201354585952641627929723523x^{8}yz^{12}+9955116052097214265382209887592785743509xy^{8}z^{12}+6796044054692944890381705389565665980173y^{9}z^{12}+3655505359651946137849081303097151788x^{8}z^{13}-224215851009248310393417737643720208242x^{7}yz^{13}+570231658595032048112841982798597981786xy^{7}z^{13}+13714392146719990642221329097830030943145y^{8}z^{13}+1511987555486014339415177503017893601526x^{7}z^{14}-3941493852566640303654747343782639607213x^{6}yz^{14}-3720206808796651952132710319811305591223xy^{6}z^{14}+9633043952077867761624309261651722495778y^{7}z^{14}+3612323339664173281920065173289068061271x^{6}z^{15}-6783620982399142653219534994116272951616x^{5}yz^{15}-5346670759180844726864688881654535625779xy^{5}z^{15}-34201372537852934041508732501710505111y^{6}z^{15}-4961564899065963814941364132053439007045x^{5}z^{16}+7930310724358849759295058134053003548101x^{4}yz^{16}-38913343755808484309731757069521292658367xy^{4}z^{16}+6853746460190606949454565936395699484664y^{5}z^{16}-10569335951735930671454397712493270778396x^{4}z^{17}+28989190410363637205798232924392600348484x^{3}yz^{17}-15389580243914860185896669165892280159498xy^{3}z^{17}-5934169730917255173078804903669511064184y^{4}z^{17}-7774152631757491372501978914563034138616x^{3}z^{18}-8200953544490131570735940376381580409555x^{2}yz^{18}+7449515000086381620660211761859766226539xy^{2}z^{18}+12353920512806658732367952500182581409655y^{3}z^{18}+6241310924390350241997074811831382917004x^{2}z^{19}-1987273160207436749624151915734713503680xyz^{19}+11559278678000816229684029932983898335543y^{2}z^{19}+24095922439220300610779554427835750610548xz^{20}-12038054179386684462807122608048029698874yz^{20}+597813843102768190422181968482304000z^{21}}{2626222354592253497209841473x^{21}-56093179572800047871290532017x^{20}y+14596543425002713109322650065628xy^{20}-11203748675703644709518698506y^{21}+468056367293055524442303584127x^{20}z-2263494866149023984969592614924x^{19}yz-193545135233842590897101344316936xy^{19}z+14655713857308983691967335288872y^{20}z+3599005360098339907833549002165x^{19}z^{2}-1180832933919691819503139503621x^{18}yz^{2}+994829590037354292022293067954202xy^{18}z^{2}-202589350240195799311140182766424y^{19}z^{2}-17436315644760433026972454884474x^{18}z^{3}+8506621252435941047520979181391x^{17}yz^{3}-1595824605392584942849257217912677xy^{17}z^{3}+1224684419035056509513056279234843y^{18}z^{3}+49923691514449958852531496122237x^{17}z^{4}+13887516762971308055434585536525x^{16}yz^{4}-2583095192167071887001250903510216xy^{16}z^{4}-1376888377925397227606015771575836y^{17}z^{4}-151168736506263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|
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
