Invariants
Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $6^{3}\cdot18^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 18D4 |
Level structure
$\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}17&0\\20&19\end{bmatrix}$, $\begin{bmatrix}41&46\\38&39\end{bmatrix}$, $\begin{bmatrix}58&3\\49&68\end{bmatrix}$, $\begin{bmatrix}70&15\\27&10\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 72.72.4.a.1 for the level structure with $-I$) |
Cyclic 72-isogeny field degree: | $12$ |
Cyclic 72-torsion field degree: | $288$ |
Full 72-torsion field degree: | $41472$ |
Rational points
This modular curve has 2 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 |
---|---|---|---|---|---|
24.48.0-24.p.1.4 | $24$ | $3$ | $3$ | $0$ | $0$ |
36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ |
72.72.2-18.c.1.16 | $72$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
72.288.9-72.i.1.3 | $72$ | $2$ | $2$ | $9$ |
72.288.9-72.j.1.10 | $72$ | $2$ | $2$ | $9$ |
72.288.9-72.m.1.1 | $72$ | $2$ | $2$ | $9$ |
72.288.9-72.n.1.6 | $72$ | $2$ | $2$ | $9$ |
72.288.9-72.q.1.4 | $72$ | $2$ | $2$ | $9$ |
72.288.9-72.r.1.8 | $72$ | $2$ | $2$ | $9$ |
72.288.9-72.s.1.4 | $72$ | $2$ | $2$ | $9$ |
72.288.9-72.t.1.11 | $72$ | $2$ | $2$ | $9$ |
72.432.10-72.a.1.16 | $72$ | $3$ | $3$ | $10$ |
72.432.10-72.b.1.14 | $72$ | $3$ | $3$ | $10$ |
72.432.10-72.b.2.10 | $72$ | $3$ | $3$ | $10$ |
72.432.10-72.c.1.6 | $72$ | $3$ | $3$ | $10$ |