Properties

Label 72.144.4-72.a.1.4
Level $72$
Index $144$
Genus $4$
Cusps $6$
$\Q$-cusps $2$

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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 $72$ $3$ $3$ $10$