Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| 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 $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8J0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}41&88\\24&119\end{bmatrix}$, $\begin{bmatrix}49&84\\40&97\end{bmatrix}$, $\begin{bmatrix}95&52\\102&119\end{bmatrix}$, $\begin{bmatrix}103&12\\94&47\end{bmatrix}$, $\begin{bmatrix}109&96\\30&71\end{bmatrix}$, $\begin{bmatrix}119&16\\52&81\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $768$ |
| 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, including 222 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{x^{24}(x^{8}-32x^{6}y^{2}+1280x^{4}y^{4}-16384x^{2}y^{6}+65536y^{8})^{3}}{y^{4}x^{32}(x-4y)^{4}(x+4y)^{4}(x^{2}-8y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.24.0-4.b.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.24.0-4.b.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.96.0-8.b.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-8.c.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-8.e.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-8.f.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-8.h.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-8.i.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.i.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.j.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-8.k.2.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.k.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-8.l.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.l.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.m.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.n.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.o.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.p.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.r.2.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.s.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.s.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.t.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.v.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.w.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.w.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.x.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.be.1.18 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bg.1.30 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bm.1.26 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bo.1.18 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bu.2.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bw.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cc.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ce.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-8.i.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-8.k.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-8.m.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-8.n.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.be.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.be.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bf.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bf.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bi.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bi.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bj.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bj.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dx.1.18 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dz.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ef.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.eh.1.18 | $120$ | $2$ | $2$ | $1$ |
| 120.144.4-24.z.1.20 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-24.bq.1.36 | $120$ | $4$ | $4$ | $3$ |
| 120.240.8-40.n.1.23 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-40.v.1.31 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-40.z.1.44 | $120$ | $10$ | $10$ | $15$ |