| $\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}11&8\\4&15\end{bmatrix}$, $\begin{bmatrix}11&30\\32&31\end{bmatrix}$, $\begin{bmatrix}17&0\\20&27\end{bmatrix}$, $\begin{bmatrix}29&12\\4&25\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
40.192.1-40.bm.2.1, 40.192.1-40.bm.2.2, 40.192.1-40.bm.2.3, 40.192.1-40.bm.2.4, 40.192.1-40.bm.2.5, 40.192.1-40.bm.2.6, 40.192.1-40.bm.2.7, 40.192.1-40.bm.2.8, 120.192.1-40.bm.2.1, 120.192.1-40.bm.2.2, 120.192.1-40.bm.2.3, 120.192.1-40.bm.2.4, 120.192.1-40.bm.2.5, 120.192.1-40.bm.2.6, 120.192.1-40.bm.2.7, 120.192.1-40.bm.2.8, 280.192.1-40.bm.2.1, 280.192.1-40.bm.2.2, 280.192.1-40.bm.2.3, 280.192.1-40.bm.2.4, 280.192.1-40.bm.2.5, 280.192.1-40.bm.2.6, 280.192.1-40.bm.2.7, 280.192.1-40.bm.2.8 |
| Cyclic 40-isogeny field degree: |
$12$ |
| Cyclic 40-torsion field degree: |
$96$ |
| Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + x y + 2 x z - 2 y^{2} + 2 y z + 2 z^{2} $ |
| $=$ | $11 x^{2} - 3 x y - 6 x z - 4 y^{2} - 6 y z - 6 z^{2} - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} y + 40 x^{2} z^{2} - 8 x y^{3} + 80 x y z^{2} - y^{4} - 80 y^{2} z^{2} + 900 z^{4} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{20}w$ |
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{2}{5}\cdot\frac{13050469826450997667430400000000000xz^{23}-1423050873904453499289600000000000xz^{21}w^{2}+63602944505485830979584000000000xz^{19}w^{4}-1561794847803799160422400000000xz^{17}w^{6}+23503106872226694103040000000xz^{15}w^{8}-226941898388722814976000000xz^{13}w^{10}+1422468858097999052800000xz^{11}w^{12}-5701726260506439680000xz^{9}w^{14}+13982081641331712000xz^{7}w^{16}-19201538693644800xz^{5}w^{18}+12373681697280xz^{3}w^{20}-2424959424xzw^{22}-11318879903251354877952000000000000y^{2}z^{22}+856437082360846721925120000000000y^{2}z^{20}w^{2}-27965518685651508830208000000000y^{2}z^{18}w^{4}+515980384059141091328000000000y^{2}z^{16}w^{6}-5915104351903089868800000000y^{2}z^{14}w^{8}+43633763623374093312000000y^{2}z^{12}w^{10}-207380377491204812800000y^{2}z^{10}w^{12}+617840927437455360000y^{2}z^{8}w^{14}-1083295033073152000y^{2}z^{6}w^{16}+987542431536000y^{2}z^{4}w^{18}-358643645760y^{2}z^{2}w^{20}+22256928y^{2}w^{22}+14720917306245079508582400000000000yz^{23}-1543562904709421688422400000000000yz^{21}w^{2}+67327234390385958567936000000000yz^{19}w^{4}-1626220684534070232678400000000yz^{17}w^{6}+24187161555658679705600000000yz^{15}w^{8}-231539377505874665472000000yz^{13}w^{10}+1441922801845090508800000yz^{11}w^{12}-5751577513475256320000yz^{9}w^{14}+14053136784399744000yz^{7}w^{16}-19248852903782400yz^{5}w^{18}+12383239090560yz^{3}w^{20}-2424959424yzw^{22}+18753734267195908540825600000000000z^{24}-284924429086431567052800000000000z^{22}w^{2}-39458485589452671983616000000000z^{20}w^{4}+1946009457618604423577600000000z^{18}w^{6}-41868035987085801651200000000z^{16}w^{8}+519904690690165929984000000z^{14}w^{10}-4023961925561781350400000z^{12}w^{12}+19729930813121740800000z^{10}w^{14}-60022820491245152000z^{8}w^{16}+106730696584339200z^{6}w^{18}-98176604608320z^{4}w^{20}+35828753472z^{2}w^{22}-2225693w^{24}}{w^{4}(51343614024152883200000000xz^{19}-4858637860832622080000000xz^{17}w^{2}+178528130825507328000000xz^{15}w^{4}-3412684927132588800000xz^{13}w^{6}+37449927078938880000xz^{11}w^{8}-241972526228864000xz^{9}w^{10}+898995915449600xz^{7}w^{12}-1775530460800xz^{5}w^{14}+1565632896xz^{3}w^{16}-397440xzw^{18}-44531132493052902400000000y^{2}z^{18}+2727629495033121920000000y^{2}z^{16}w^{2}-69257293417756800000000y^{2}z^{14}w^{4}+942655355021140800000y^{2}z^{12}w^{6}-7425450440508640000y^{2}z^{10}w^{8}+34072769212968000y^{2}z^{8}w^{10}-86940689696000y^{2}z^{6}w^{12}+109745238800y^{2}z^{4}w^{14}-52498800y^{2}z^{2}w^{16}+4050y^{2}w^{18}+57915546819731788800000000yz^{19}-5238043960584793600000000yz^{17}w^{2}+187497633607988288000000yz^{15}w^{4}-3524475156119411200000yz^{13}w^{6}+38237376654743680000yz^{11}w^{8}-245090093420096000yz^{9}w^{10}+905467223737600yz^{7}w^{12}-1781475786880yz^{5}w^{14}+1567196736yz^{3}w^{16}-397440yzw^{18}+73781596105822912000000000z^{20}-57598573016718080000000z^{18}w^{2}-158478583571470152000000z^{16}w^{4}+5374067606228368000000z^{14}w^{6}-82077936490992000000z^{12}w^{8}+686808660135696000z^{10}w^{10}-3265627152600000z^{8}w^{12}+8516125075200z^{6}w^{14}-10889902856z^{4}w^{16}+5243400z^{2}w^{18}-405w^{20})}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.
