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Properties

Label 120.12.0.s.1
Level $120$
Index $12$
Genus $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $120$ $\SL_2$-level: $4$
Index: $12$ $\PSL_2$-index:$12$
Genus: $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $2^{2}\cdot4^{2}$ Cusp orbits $1^{2}\cdot2$
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: 4E0

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}1&52\\90&119\end{bmatrix}$, $\begin{bmatrix}7&52\\65&99\end{bmatrix}$, $\begin{bmatrix}39&88\\88&13\end{bmatrix}$, $\begin{bmatrix}49&28\\35&109\end{bmatrix}$, $\begin{bmatrix}97&92\\110&107\end{bmatrix}$, $\begin{bmatrix}117&112\\118&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 120.24.0-120.s.1.1, 120.24.0-120.s.1.2, 120.24.0-120.s.1.3, 120.24.0-120.s.1.4, 120.24.0-120.s.1.5, 120.24.0-120.s.1.6, 120.24.0-120.s.1.7, 120.24.0-120.s.1.8, 120.24.0-120.s.1.9, 120.24.0-120.s.1.10, 120.24.0-120.s.1.11, 120.24.0-120.s.1.12, 120.24.0-120.s.1.13, 120.24.0-120.s.1.14, 120.24.0-120.s.1.15, 120.24.0-120.s.1.16
Cyclic 120-isogeny field degree: $48$
Cyclic 120-torsion field degree: $1536$
Full 120-torsion field degree: $2949120$

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
$X_0(4)$ $4$ $2$ $2$ $0$ $0$
120.6.0.a.1 $120$ $2$ $2$ $0$ $?$
120.6.0.e.1 $120$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
120.24.0.de.1 $120$ $2$ $2$ $0$
120.24.0.df.1 $120$ $2$ $2$ $0$
120.24.0.dq.1 $120$ $2$ $2$ $0$
120.24.0.dr.1 $120$ $2$ $2$ $0$
120.24.0.du.1 $120$ $2$ $2$ $0$
120.24.0.dv.1 $120$ $2$ $2$ $0$
120.24.0.dy.1 $120$ $2$ $2$ $0$
120.24.0.dz.1 $120$ $2$ $2$ $0$
120.36.2.co.1 $120$ $3$ $3$ $2$
120.48.1.zo.1 $120$ $4$ $4$ $1$
120.60.4.bq.1 $120$ $5$ $5$ $4$
120.72.3.bgs.1 $120$ $6$ $6$ $3$
120.120.7.co.1 $120$ $10$ $10$ $7$