Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2^{2}\cdot5^{2}\cdot10^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.0.4 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}15&6\\16&25\end{bmatrix}$, $\begin{bmatrix}17&29\\16&15\end{bmatrix}$, $\begin{bmatrix}19&7\\22&19\end{bmatrix}$, $\begin{bmatrix}29&5\\26&33\end{bmatrix}$, $\begin{bmatrix}35&4\\8&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 10.36.0.a.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $10240$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 42 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^5}\cdot\frac{x^{36}(x^{12}+32x^{11}y+416x^{10}y^{2}+2880x^{9}y^{3}+11520x^{8}y^{4}+18432x^{7}y^{5}-65536x^{6}y^{6}-442368x^{5}y^{7}-983040x^{4}y^{8}-655360x^{3}y^{9}+1048576x^{2}y^{10}+2097152xy^{11}+1048576y^{12})^{3}}{y^{5}x^{46}(x+2y)^{5}(x+4y)^{10}(x^{2}+2xy-4y^{2})(x^{2}+12xy+16y^{2})^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.