Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x y + x z + y^{2} + z^{2} + w^{2} $ |
| $=$ | $2 x^{2} - x y + x z - y^{2} - 4 y z - z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7 x^{4} + 24 x^{3} y + 38 x^{2} y^{2} + 11 x^{2} z^{2} + 24 x y^{3} + 18 x y z^{2} + 7 y^{4} + \cdots + 4 z^{4} $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{7^4}\cdot\frac{41582320707396157440xz^{23}+90546041818686455808xz^{21}w^{2}-35310370883523649536xz^{19}w^{4}-937760888087153500160xz^{17}w^{6}-2579182757072217081856xz^{15}w^{8}-3954604772401025433600xz^{13}w^{10}-3680163736364897739776xz^{11}w^{12}-2027207595887681514496xz^{9}w^{14}-510717745471684376640xz^{7}w^{16}-3493789682478827648xz^{5}w^{18}+25583842574599277024xz^{3}w^{20}+3459113231027054496xzw^{22}+51074184642234777600y^{2}z^{22}-27001727176555192320y^{2}z^{20}w^{2}-677898040864708780032y^{2}z^{18}w^{4}-2622069142819421216768y^{2}z^{16}w^{6}-4946413302590120374272y^{2}z^{14}w^{8}-5483264120449823984640y^{2}z^{12}w^{10}-3124013010453619622912y^{2}z^{10}w^{12}-69213074684791468032y^{2}z^{8}w^{14}+1115904600065660598144y^{2}z^{6}w^{16}+591312026417366314144y^{2}z^{4}w^{18}+121854342787455342624y^{2}z^{2}w^{20}+9118177408472621664y^{2}w^{22}+105087332786679644160yz^{23}+89191284197373739008yz^{21}w^{2}-730525989170920341504yz^{19}w^{4}-3864254520041529040896yz^{17}w^{6}-8228122997128184700928yz^{15}w^{8}-10326656903066850447360yz^{13}w^{10}-7262708342882999494656yz^{11}w^{12}-1978029517026405520384yz^{9}w^{14}+968014580192647792896yz^{7}w^{16}+780767548646198322048yz^{5}w^{18}+182219056246717163072yz^{3}w^{20}+14777241585918188832yzw^{22}+77154427810840211456z^{24}+146934674525117054976z^{22}w^{2}-301260284150899777536z^{20}w^{4}-2618312371643652009984z^{18}w^{6}-6833144857309039067904z^{16}w^{8}-10268716759657585754112z^{14}w^{10}-9391703485719184809984z^{12}w^{12}-4704066538096129602048z^{10}w^{14}-433248091275448904112z^{8}w^{16}+808895979605199541888z^{6}w^{18}+405786382532043905952z^{4}w^{20}+77113840065536596560z^{2}w^{22}+5208356359889633615w^{24}}{w^{4}(36807283461920xz^{19}-336292642352096xz^{17}w^{2}-970891549605656xz^{15}w^{4}-766689388119872xz^{13}w^{6}-212299491224744xz^{11}w^{8}-16878887687928xz^{9}w^{10}+405786191160xz^{7}w^{12}+84903989128xz^{5}w^{14}-7108823176xz^{3}w^{16}-4980788064xzw^{18}-82325094506880y^{2}z^{18}-1056916056667856y^{2}z^{16}w^{2}-1799162174967696y^{2}z^{14}w^{4}-756751366892612y^{2}z^{12}w^{6}+64222398744532y^{2}z^{10}w^{8}+47884283892774y^{2}z^{8}w^{10}+417977720048y^{2}z^{6}w^{12}+176035610422y^{2}z^{4}w^{14}+21237526884y^{2}z^{2}w^{16}-1089547389y^{2}w^{18}-22543828453376yz^{19}-1491212581076544yz^{17}w^{2}-3074524799936352yz^{15}w^{4}-1641897391062320yz^{13}w^{6}-75938061581592yz^{11}w^{8}+59013552041276yz^{9}w^{10}+489148104424yz^{7}w^{12}+285772715172yz^{5}w^{14}+51935915752yz^{3}w^{16}+2801693286yzw^{18}+22470507657840z^{20}-893839904358560z^{18}w^{2}-2581438647981809z^{16}w^{4}-2351056902944288z^{14}w^{6}-716888305076510z^{12}w^{8}+23073980325438z^{10}w^{10}+28370587955717z^{8}w^{12}+749933079574z^{6}w^{14}+233644090358z^{4}w^{16}+23676037707z^{2}w^{18}-1089547389w^{20})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.1.s.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 7X^{4}+24X^{3}Y+38X^{2}Y^{2}+24XY^{3}+7Y^{4}+11X^{2}Z^{2}+18XYZ^{2}+11Y^{2}Z^{2}+4Z^{4} $ |
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.
