Invariants
| Level: | $40$ | $\SL_2$-level: | $4$ | ||||
| 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 | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.35 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&14\\32&9\end{bmatrix}$, $\begin{bmatrix}23&18\\17&21\end{bmatrix}$, $\begin{bmatrix}29&12\\26&1\end{bmatrix}$, $\begin{bmatrix}31&4\\16&11\end{bmatrix}$, $\begin{bmatrix}35&2\\13&37\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $61440$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(x+4y)^{12}(5x^{2}-96y^{2})^{3}(15x^{2}-32y^{2})^{3}}{(x+4y)^{12}(5x^{2}-32y^{2})^{2}(5x^{2}+32y^{2})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 40.6.0.a.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.6.0.c.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.24.0.ce.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cf.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cm.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cn.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.co.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cp.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cs.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ct.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.1.z.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bb.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ej.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.el.1 | $40$ | $2$ | $2$ | $1$ |
| 40.60.4.ca.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.ea.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.ga.1 | $40$ | $10$ | $10$ | $7$ |
| 120.24.0.fw.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fx.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ge.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gf.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gg.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gh.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gk.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gl.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.ef.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.eh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ph.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pj.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.2.qs.1 | $120$ | $3$ | $3$ | $2$ |
| 120.48.1.bzq.1 | $120$ | $4$ | $4$ | $1$ |
| 280.24.0.em.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.en.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.eq.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.er.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.es.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.et.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ew.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ex.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.de.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.df.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.fq.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.fr.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.5.fg.1 | $280$ | $8$ | $8$ | $5$ |
| 280.252.16.ke.1 | $280$ | $21$ | $21$ | $16$ |
| 280.336.21.ke.1 | $280$ | $28$ | $28$ | $21$ |