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: | 40.12.0.36 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&4\\8&11\end{bmatrix}$, $\begin{bmatrix}3&16\\25&17\end{bmatrix}$, $\begin{bmatrix}3&24\\14&31\end{bmatrix}$, $\begin{bmatrix}9&16\\36&37\end{bmatrix}$, $\begin{bmatrix}35&14\\19&5\end{bmatrix}$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(x-4y)^{12}(5x^{2}+96y^{2})^{3}(15x^{2}+32y^{2})^{3}}{(x-4y)^{12}(5x^{2}-32y^{2})^{4}(5x^{2}+32y^{2})^{2}}$ |
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.b.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.6.0.d.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.ci.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cj.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cw.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cx.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cy.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cz.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dc.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dd.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.1.bd.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bf.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.en.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ep.1 | $40$ | $2$ | $2$ | $1$ |
| 40.60.4.cd.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.ed.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.gd.1 | $40$ | $10$ | $10$ | $7$ |
| 120.24.0.ga.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gb.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.go.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gp.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gq.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gu.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gv.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.ej.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.el.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pl.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pn.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.2.qv.1 | $120$ | $3$ | $3$ | $2$ |
| 120.48.1.bzt.1 | $120$ | $4$ | $4$ | $1$ |
| 280.24.0.fa.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fb.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fe.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ff.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fg.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fh.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fk.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fl.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.di.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.dj.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.fu.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.fv.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.5.fj.1 | $280$ | $8$ | $8$ | $5$ |
| 280.252.16.kh.1 | $280$ | $21$ | $21$ | $16$ |
| 280.336.21.kh.1 | $280$ | $28$ | $28$ | $21$ |