Invariants
Level: | $72$ | $\SL_2$-level: | $72$ | Newform level: | $1$ | ||
Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $1^{6}\cdot2^{3}\cdot8^{3}\cdot9^{2}\cdot18\cdot72$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 72B5 |
Level structure
$\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}16&39\\21&70\end{bmatrix}$, $\begin{bmatrix}20&17\\9&52\end{bmatrix}$, $\begin{bmatrix}40&61\\53&48\end{bmatrix}$, $\begin{bmatrix}55&12\\20&35\end{bmatrix}$, $\begin{bmatrix}58&55\\57&32\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 72.144.5.br.1 for the level structure with $-I$) |
Cyclic 72-isogeny field degree: | $2$ |
Cyclic 72-torsion field degree: | $24$ |
Full 72-torsion field degree: | $20736$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.p.1.7 | $8$ | $12$ | $12$ | $0$ | $0$ |
$X_0(9)$ | $9$ | $24$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.96.1-24.ix.1.22 | $24$ | $3$ | $3$ | $1$ | $0$ |
72.144.1-36.c.1.5 | $72$ | $2$ | $2$ | $1$ | $?$ |
72.144.1-36.c.1.12 | $72$ | $2$ | $2$ | $1$ | $?$ |