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Properties

Label 120.24.0.df.1
Level $120$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $120$ $\SL_2$-level: $8$
Index: $24$ $\PSL_2$-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 $2^{4}\cdot8^{2}$ 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: 8G0

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}17&96\\103&67\end{bmatrix}$, $\begin{bmatrix}41&40\\82&69\end{bmatrix}$, $\begin{bmatrix}63&28\\20&81\end{bmatrix}$, $\begin{bmatrix}93&8\\41&67\end{bmatrix}$, $\begin{bmatrix}97&108\\65&73\end{bmatrix}$, $\begin{bmatrix}113&12\\5&49\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 120.48.0-120.df.1.1, 120.48.0-120.df.1.2, 120.48.0-120.df.1.3, 120.48.0-120.df.1.4, 120.48.0-120.df.1.5, 120.48.0-120.df.1.6, 120.48.0-120.df.1.7, 120.48.0-120.df.1.8, 120.48.0-120.df.1.9, 120.48.0-120.df.1.10, 120.48.0-120.df.1.11, 120.48.0-120.df.1.12, 120.48.0-120.df.1.13, 120.48.0-120.df.1.14, 120.48.0-120.df.1.15, 120.48.0-120.df.1.16, 120.48.0-120.df.1.17, 120.48.0-120.df.1.18, 120.48.0-120.df.1.19, 120.48.0-120.df.1.20, 120.48.0-120.df.1.21, 120.48.0-120.df.1.22, 120.48.0-120.df.1.23, 120.48.0-120.df.1.24, 240.48.0-120.df.1.1, 240.48.0-120.df.1.2, 240.48.0-120.df.1.3, 240.48.0-120.df.1.4, 240.48.0-120.df.1.5, 240.48.0-120.df.1.6, 240.48.0-120.df.1.7, 240.48.0-120.df.1.8, 240.48.0-120.df.1.9, 240.48.0-120.df.1.10, 240.48.0-120.df.1.11, 240.48.0-120.df.1.12, 240.48.0-120.df.1.13, 240.48.0-120.df.1.14, 240.48.0-120.df.1.15, 240.48.0-120.df.1.16
Cyclic 120-isogeny field degree: $24$
Cyclic 120-torsion field degree: $768$
Full 120-torsion field degree: $1474560$

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
$X_0(8)$ $8$ $2$ $2$ $0$ $0$
120.12.0.s.1 $120$ $2$ $2$ $0$ $?$
120.12.0.bb.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
120.48.0.eh.1 $120$ $2$ $2$ $0$
120.48.0.eh.2 $120$ $2$ $2$ $0$
120.48.0.ei.1 $120$ $2$ $2$ $0$
120.48.0.ei.2 $120$ $2$ $2$ $0$
120.48.0.ej.1 $120$ $2$ $2$ $0$
120.48.0.ej.2 $120$ $2$ $2$ $0$
120.48.0.ek.1 $120$ $2$ $2$ $0$
120.48.0.ek.2 $120$ $2$ $2$ $0$
120.72.4.md.1 $120$ $3$ $3$ $4$
120.96.3.pl.1 $120$ $4$ $4$ $3$
120.120.8.fd.1 $120$ $5$ $5$ $8$
120.144.7.dvq.1 $120$ $6$ $6$ $7$
120.240.15.lv.1 $120$ $10$ $10$ $15$
240.48.0.bw.1 $240$ $2$ $2$ $0$
240.48.0.bw.2 $240$ $2$ $2$ $0$
240.48.0.bx.1 $240$ $2$ $2$ $0$
240.48.0.bx.2 $240$ $2$ $2$ $0$
240.48.0.by.1 $240$ $2$ $2$ $0$
240.48.0.by.2 $240$ $2$ $2$ $0$
240.48.0.bz.1 $240$ $2$ $2$ $0$
240.48.0.bz.2 $240$ $2$ $2$ $0$
240.48.1.r.1 $240$ $2$ $2$ $1$
240.48.1.t.1 $240$ $2$ $2$ $1$
240.48.1.ec.1 $240$ $2$ $2$ $1$
240.48.1.ef.1 $240$ $2$ $2$ $1$
240.48.1.he.1 $240$ $2$ $2$ $1$
240.48.1.hh.1 $240$ $2$ $2$ $1$
240.48.1.hv.1 $240$ $2$ $2$ $1$
240.48.1.hx.1 $240$ $2$ $2$ $1$