Properties

Label 120.48.0-120.e.1.9
Level $120$
Index $48$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $120$ $\SL_2$-level: $4$
Index: $48$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $4^{6}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 4G0

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}33&94\\52&79\end{bmatrix}$, $\begin{bmatrix}37&48\\76&83\end{bmatrix}$, $\begin{bmatrix}39&74\\92&71\end{bmatrix}$, $\begin{bmatrix}47&16\\76&1\end{bmatrix}$, $\begin{bmatrix}47&32\\8&75\end{bmatrix}$, $\begin{bmatrix}71&104\\80&13\end{bmatrix}$
Contains $-I$: no $\quad$ (see 120.24.0.e.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $48$
Cyclic 120-torsion field degree: $1536$
Full 120-torsion field degree: $737280$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
12.24.0-4.b.1.1 $12$ $2$ $2$ $0$ $0$
40.24.0-4.b.1.6 $40$ $2$ $2$ $0$ $0$
120.24.0-120.a.1.6 $120$ $2$ $2$ $0$ $?$
120.24.0-120.a.1.14 $120$ $2$ $2$ $0$ $?$
120.24.0-120.b.1.6 $120$ $2$ $2$ $0$ $?$
120.24.0-120.b.1.14 $120$ $2$ $2$ $0$ $?$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
120.96.0-120.bh.1.5 $120$ $2$ $2$ $0$
120.96.0-120.bh.2.7 $120$ $2$ $2$ $0$
120.96.0-120.bi.1.11 $120$ $2$ $2$ $0$
120.96.0-120.bi.2.15 $120$ $2$ $2$ $0$
120.96.0-120.bj.1.10 $120$ $2$ $2$ $0$
120.96.0-120.bj.2.17 $120$ $2$ $2$ $0$
120.96.0-120.bk.1.6 $120$ $2$ $2$ $0$
120.96.0-120.bk.2.3 $120$ $2$ $2$ $0$
120.96.0-120.bl.1.12 $120$ $2$ $2$ $0$
120.96.0-120.bl.2.1 $120$ $2$ $2$ $0$
120.96.0-120.bm.1.5 $120$ $2$ $2$ $0$
120.96.0-120.bm.2.18 $120$ $2$ $2$ $0$
120.96.0-120.bn.1.10 $120$ $2$ $2$ $0$
120.96.0-120.bn.2.12 $120$ $2$ $2$ $0$
120.96.0-120.bo.1.5 $120$ $2$ $2$ $0$
120.96.0-120.bo.2.6 $120$ $2$ $2$ $0$
120.96.1-120.r.1.20 $120$ $2$ $2$ $1$
120.96.1-120.ba.1.20 $120$ $2$ $2$ $1$
120.96.1-120.cz.1.8 $120$ $2$ $2$ $1$
120.96.1-120.de.1.8 $120$ $2$ $2$ $1$
120.96.1-120.ez.1.8 $120$ $2$ $2$ $1$
120.96.1-120.fe.1.8 $120$ $2$ $2$ $1$
120.96.1-120.fp.1.20 $120$ $2$ $2$ $1$
120.96.1-120.fr.1.20 $120$ $2$ $2$ $1$
120.144.4-120.l.1.18 $120$ $3$ $3$ $4$
120.192.3-120.dz.1.58 $120$ $4$ $4$ $3$
120.240.8-120.l.1.13 $120$ $5$ $5$ $8$
120.288.7-120.ei.1.1 $120$ $6$ $6$ $7$
120.480.15-120.l.1.24 $120$ $10$ $10$ $15$