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: | 40.72.1.194 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&16\\8&29\end{bmatrix}$, $\begin{bmatrix}5&11\\22&19\end{bmatrix}$, $\begin{bmatrix}13&23\\22&9\end{bmatrix}$, $\begin{bmatrix}21&10\\34&37\end{bmatrix}$ |
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 $ | $=$ | $ 2 x^{2} - z^{2} - z w $ |
$=$ | $4 x^{2} + 10 y^{2} + 2 z^{2} + 6 z w + 5 w^{2}$ |
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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{(16z^{6}+80z^{5}w+160z^{4}w^{2}+160z^{3}w^{3}+80z^{2}w^{4}+20zw^{5}+5w^{6})^{3}}{w^{5}z^{2}(z+w)^{10}(4z+5w)}$ |
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.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.d.2 | $40$ | $2$ | $2$ | $0$ | $0$ | 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.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.bl.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.cs.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.cw.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.ij.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.ik.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.jb.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.jc.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.360.13.ca.1 | $40$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
120.144.5.chc.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.chg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cie.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cii.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egr.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egu.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eht.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehw.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.ui.1 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.iem.1 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.bk.1 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.bfs.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bfu.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgg.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgi.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.boj.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bok.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.box.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.boy.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |