Invariants
Level: | $40$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.166 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}23&16\\9&39\end{bmatrix}$, $\begin{bmatrix}27&4\\34&9\end{bmatrix}$, $\begin{bmatrix}29&4\\25&13\end{bmatrix}$, $\begin{bmatrix}33&28\\8&25\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 20.12.0.h.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $30720$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 480 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{5}\cdot\frac{x^{12}(11x^{4}+192x^{3}y-4736x^{2}y^{2}+12288xy^{3}+45056y^{4})^{3}}{x^{12}(x-8y)^{2}(x+8y)^{2}(3x^{2}-32xy+192y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.12.0-4.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
40.12.0-4.c.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.