Invariants
Level: | $40$ | $\SL_2$-level: | $5$ | ||||
Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $1^{2}\cdot5^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 5D0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.44 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&15\\30&21\end{bmatrix}$, $\begin{bmatrix}2&35\\33&19\end{bmatrix}$, $\begin{bmatrix}11&3\\18&1\end{bmatrix}$, $\begin{bmatrix}34&39\\37&36\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $61440$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 2 x^{2} + 5 y^{2} - 100 y z + 520 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
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 |
---|---|---|---|---|---|
$X_0(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
40.24.1.bw.2 | $40$ | $2$ | $2$ | $1$ |
40.24.1.by.2 | $40$ | $2$ | $2$ | $1$ |
40.24.1.cc.2 | $40$ | $2$ | $2$ | $1$ |
40.24.1.ce.2 | $40$ | $2$ | $2$ | $1$ |
40.24.1.ci.2 | $40$ | $2$ | $2$ | $1$ |
40.24.1.ck.1 | $40$ | $2$ | $2$ | $1$ |
40.24.1.co.2 | $40$ | $2$ | $2$ | $1$ |
40.24.1.cq.2 | $40$ | $2$ | $2$ | $1$ |
40.36.0.a.1 | $40$ | $3$ | $3$ | $0$ |
40.48.3.i.2 | $40$ | $4$ | $4$ | $3$ |
40.60.0.a.1 | $40$ | $5$ | $5$ | $0$ |
120.24.1.em.2 | $120$ | $2$ | $2$ | $1$ |
120.24.1.eo.1 | $120$ | $2$ | $2$ | $1$ |
120.24.1.es.2 | $120$ | $2$ | $2$ | $1$ |
120.24.1.eu.2 | $120$ | $2$ | $2$ | $1$ |
120.24.1.jk.2 | $120$ | $2$ | $2$ | $1$ |
120.24.1.jm.2 | $120$ | $2$ | $2$ | $1$ |
120.24.1.jq.2 | $120$ | $2$ | $2$ | $1$ |
120.24.1.js.2 | $120$ | $2$ | $2$ | $1$ |
120.36.2.rk.1 | $120$ | $3$ | $3$ | $2$ |
120.48.1.cag.1 | $120$ | $4$ | $4$ | $1$ |
200.60.0.a.2 | $200$ | $5$ | $5$ | $0$ |
280.24.1.gn.2 | $280$ | $2$ | $2$ | $1$ |
280.24.1.go.1 | $280$ | $2$ | $2$ | $1$ |
280.24.1.gq.2 | $280$ | $2$ | $2$ | $1$ |
280.24.1.gr.2 | $280$ | $2$ | $2$ | $1$ |
280.24.1.gz.2 | $280$ | $2$ | $2$ | $1$ |
280.24.1.ha.2 | $280$ | $2$ | $2$ | $1$ |
280.24.1.hc.2 | $280$ | $2$ | $2$ | $1$ |
280.24.1.hd.2 | $280$ | $2$ | $2$ | $1$ |
280.96.5.gi.1 | $280$ | $8$ | $8$ | $5$ |
280.252.16.ki.1 | $280$ | $21$ | $21$ | $16$ |
280.336.21.ki.2 | $280$ | $28$ | $28$ | $21$ |