Invariants
Level: | $72$ | $\SL_2$-level: | $72$ | Newform level: | $1$ | ||
Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $3^{2}\cdot6\cdot9^{2}\cdot18\cdot24\cdot72$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $4 \le \gamma \le 9$ | ||||||
$\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 72E9 |
Level structure
$\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}12&41\\55&38\end{bmatrix}$, $\begin{bmatrix}21&20\\14&27\end{bmatrix}$, $\begin{bmatrix}29&46\\42&61\end{bmatrix}$, $\begin{bmatrix}34&17\\49&6\end{bmatrix}$, $\begin{bmatrix}60&13\\1&0\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 72.144.9.dc.1 for the level structure with $-I$) |
Cyclic 72-isogeny field degree: | $6$ |
Cyclic 72-torsion field degree: | $72$ |
Full 72-torsion field degree: | $20736$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.96.1-24.iu.1.18 | $24$ | $3$ | $3$ | $1$ | $0$ |
36.144.4-36.d.1.4 | $36$ | $2$ | $2$ | $4$ | $0$ |
72.144.4-36.d.1.1 | $72$ | $2$ | $2$ | $4$ | $?$ |