Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 20H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.193 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&12\\34&15\end{bmatrix}$, $\begin{bmatrix}11&3\\22&7\end{bmatrix}$, $\begin{bmatrix}11&21\\32&5\end{bmatrix}$, $\begin{bmatrix}35&21\\24&27\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $4$ |
Cyclic 40-torsion field degree: | $64$ |
Full 40-torsion field degree: | $10240$ |
Jacobian
Conductor: | $2^{4}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 400.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - 2 x z + 5 z^{2} + 2 w^{2} $ |
$=$ | $ - 5 x z + 2 y^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} - 4 x^{2} z^{2} + 10 y^{2} z^{2} + 4 z^{4} $ |
Rational points
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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{5}{2}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{216xz^{17}+432xz^{15}w^{2}-360xz^{13}w^{4}-1768xz^{11}w^{6}-2120xz^{9}w^{8}-1296xz^{7}w^{10}-442xz^{5}w^{12}-80xz^{3}w^{14}-6xzw^{16}+2160z^{18}+11664z^{16}w^{2}+25920z^{14}w^{4}+31036z^{12}w^{6}+21992z^{10}w^{8}+9480z^{8}w^{10}+2416z^{6}w^{12}+320z^{4}w^{14}+12z^{2}w^{16}-w^{18}}{z^{10}(2z^{2}+w^{2})^{2}(2xz^{3}+2xzw^{2}+20z^{4}+13z^{2}w^{2}+2w^{4})}$ |
Modular covers
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.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.36.1.e.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.36.0.a.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.d.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.144.5.n.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.bl.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.cs.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.cw.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.ij.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.ik.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.jb.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.jc.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.360.13.ca.1 | $40$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
120.144.5.chc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.chg.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cie.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cii.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eht.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.ui.2 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.iem.2 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.bk.2 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.bfs.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bfu.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgg.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgi.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.boj.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bok.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.box.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.boy.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |