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 $2$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}9&32\\86&91\end{bmatrix}$, $\begin{bmatrix}39&16\\14&105\end{bmatrix}$, $\begin{bmatrix}63&88\\56&9\end{bmatrix}$, $\begin{bmatrix}69&100\\2&47\end{bmatrix}$, $\begin{bmatrix}83&76\\114&65\end{bmatrix}$, $\begin{bmatrix}117&80\\110&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.i.2 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, including 90 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^4}\cdot\frac{x^{24}(81x^{8}+3456x^{6}y^{2}+184320x^{4}y^{4}+3145728x^{2}y^{6}+16777216y^{8})^{3}}{y^{4}x^{32}(3x^{2}+32y^{2})^{2}(3x^{2}+64y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 20.24.0-4.b.1.3 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 120.24.0-4.b.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.96.0-24.b.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.c.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.e.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.f.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.h.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.j.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.l.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.n.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.q.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.s.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.t.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.u.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.u.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.w.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.x.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.y.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.y.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.z.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bb.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bc.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bd.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bf.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bl.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bn.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bt.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bv.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cb.2.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cd.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ch.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ci.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cl.2.9 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cm.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.q.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.s.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.x.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.y.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bd.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bf.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bh.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bj.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dm.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dn.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dq.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dr.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dw.2.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dy.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ee.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.eg.2.3 | $120$ | $2$ | $2$ | $1$ |
| 120.144.4-24.y.2.8 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-24.bp.1.4 | $120$ | $4$ | $4$ | $3$ |
| 120.240.8-120.bd.1.58 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-120.ys.1.42 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-120.bp.1.68 | $120$ | $10$ | $10$ | $15$ |