$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}7&22\\34&1\end{bmatrix}$, $\begin{bmatrix}23&6\\30&1\end{bmatrix}$, $\begin{bmatrix}25&2\\34&31\end{bmatrix}$, $\begin{bmatrix}29&20\\16&21\end{bmatrix}$, $\begin{bmatrix}33&6\\2&3\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.b.1.1, 40.96.1-40.b.1.2, 40.96.1-40.b.1.3, 40.96.1-40.b.1.4, 40.96.1-40.b.1.5, 40.96.1-40.b.1.6, 120.96.1-40.b.1.1, 120.96.1-40.b.1.2, 120.96.1-40.b.1.3, 120.96.1-40.b.1.4, 120.96.1-40.b.1.5, 120.96.1-40.b.1.6, 280.96.1-40.b.1.1, 280.96.1-40.b.1.2, 280.96.1-40.b.1.3, 280.96.1-40.b.1.4, 280.96.1-40.b.1.5, 280.96.1-40.b.1.6 |
Cyclic 40-isogeny field degree: |
$24$ |
Cyclic 40-torsion field degree: |
$384$ |
Full 40-torsion field degree: |
$15360$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} - x y + x z - y^{2} - z^{2} $ |
| $=$ | $3 x^{2} + x y - x z + 2 y^{2} + 2 y z + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 54 x^{4} + 56 x^{3} y + 104 x^{2} y^{2} + 33 x^{2} z^{2} + 56 x y^{3} + 6 x y z^{2} + 54 y^{4} + \cdots + 4 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^6\cdot3^3}{5^3}\cdot\frac{727214687280000000xz^{11}+458442098948000000xz^{9}w^{2}-582250550354640000xz^{7}w^{4}-194435428890672000xz^{5}w^{6}-17920297233352800xz^{3}w^{8}-446962713487200xzw^{10}-349787973661600000y^{2}z^{10}-933527654029800000y^{2}z^{8}w^{2}+101598016242096000y^{2}z^{6}w^{4}+124799559111108000y^{2}z^{4}w^{6}+15264209715601140y^{2}z^{2}w^{8}-272425952034531y^{2}w^{10}-210026819523200000yz^{11}-863819491379600000yz^{9}w^{2}+36129418623792000yz^{7}w^{4}+79215178126536000yz^{5}w^{6}+6821178137765280yz^{3}w^{8}-1153183860568962yzw^{10}-520514561001600000z^{12}-1113401671642600000z^{10}w^{2}-150916519293444000z^{8}w^{4}+130045007944764000z^{6}w^{6}+49853122186926240z^{4}w^{8}+4612135987988109z^{2}w^{10}-93840990845580w^{12}}{5817717498240000xz^{11}+1169900770336000xz^{9}w^{2}-2738865878559360xz^{7}w^{4}-743665190823360xz^{5}w^{6}-118639739145600xz^{3}w^{8}-7837946959680xzw^{10}-2798303789292800y^{2}z^{10}+4265775332222400y^{2}z^{8}w^{2}+2722147531519104y^{2}z^{6}w^{4}+251490635784000y^{2}z^{4}w^{6}+18124563596724y^{2}z^{2}w^{8}-1028601398295y^{2}w^{10}-1680214556185600yz^{11}+5231458984764800yz^{9}w^{2}+2328769460020608yz^{7}w^{4}-9987418739520yz^{5}w^{6}-12718803140352yz^{3}w^{8}-3230656411050yzw^{10}-4164116488012800z^{12}+3269492233547200z^{10}w^{2}+5415676262139744z^{8}w^{4}+1432024747144704z^{6}w^{6}+126543831377904z^{4}w^{8}+6410331155529z^{2}w^{10}+534640274820w^{12}}$ |
Hi
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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.