Invariants
| Level: | $72$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{3}\cdot12^{3}$ | Cusp orbits | $3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12F2 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}30&29\\29&63\end{bmatrix}$, $\begin{bmatrix}47&13\\47&66\end{bmatrix}$, $\begin{bmatrix}55&3\\25&32\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.48.2.b.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $432$ |
| Full 72-torsion field degree: | $62208$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 18.48.0-18.a.1.2 | $18$ | $2$ | $2$ | $0$ | $0$ |
| 24.32.0-24.a.2.7 | $24$ | $3$ | $3$ | $0$ | $0$ |
| 72.48.0-18.a.1.7 | $72$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.288.7-72.i.1.5 | $72$ | $3$ | $3$ | $7$ |
| 72.288.7-72.l.1.6 | $72$ | $3$ | $3$ | $7$ |
| 72.288.10-72.o.1.6 | $72$ | $3$ | $3$ | $10$ |
| 72.288.10-72.r.1.6 | $72$ | $3$ | $3$ | $10$ |
| 72.384.5-72.b.1.6 | $72$ | $4$ | $4$ | $5$ |