Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6I0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}23&28\\96&55\end{bmatrix}$, $\begin{bmatrix}51&26\\58&107\end{bmatrix}$, $\begin{bmatrix}61&74\\24&59\end{bmatrix}$, $\begin{bmatrix}91&54\\10&53\end{bmatrix}$, $\begin{bmatrix}105&62\\98&117\end{bmatrix}$, $\begin{bmatrix}107&112\\90&13\end{bmatrix}$, $\begin{bmatrix}113&12\\30&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.24.0.a.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| 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 110 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{24}(x^{2}+12y^{2})^{3}(x^{6}-60x^{4}y^{2}+1200x^{2}y^{4}+192y^{6})^{3}}{y^{6}x^{26}(x-6y)^{2}(x-2y)^{6}(x+2y)^{6}(x+6y)^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-2.a.1.1 | $40$ | $4$ | $4$ | $0$ | $0$ |
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-2.a.1.1 | $40$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.144.1-6.a.1.7 | $120$ | $3$ | $3$ | $1$ |
| 120.96.0-12.a.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.2.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.2.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-12.a.2.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-12.a.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.a.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.b.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.b.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.b.1.27 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.c.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.c.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.c.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.d.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.d.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-12.d.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-12.a.1.1 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-12.a.1.2 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-12.a.2.1 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-12.a.2.5 | $120$ | $2$ | $2$ | $2$ |
| 120.96.0-24.o.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.1.18 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.1.26 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.10 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.24 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.32 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.bw.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bw.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bw.1.17 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bx.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bx.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bx.1.17 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.by.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.by.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.by.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bz.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bz.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bz.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-24.b.1.3 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.b.1.9 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.b.2.2 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.b.2.8 | $120$ | $2$ | $2$ | $2$ |
| 120.240.8-30.a.1.7 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-30.a.1.7 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-30.a.1.8 | $120$ | $10$ | $10$ | $15$ |
| 120.96.0-60.a.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.1.28 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.1.29 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.1.30 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.2.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.2.12 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.2.24 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.2.29 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-60.a.2.30 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-60.a.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.a.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.a.1.17 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.b.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.b.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.b.1.20 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.c.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.c.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.c.1.19 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.d.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.d.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-60.d.1.20 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-60.a.1.2 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-60.a.1.6 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-60.a.2.2 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-60.a.2.10 | $120$ | $2$ | $2$ | $2$ |
| 120.96.0-120.o.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.20 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.32 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.38 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.39 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.42 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.26 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.33 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.36 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.56 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.dg.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dg.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dg.1.38 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dh.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dh.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dh.1.29 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.di.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.di.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.di.1.36 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dj.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dj.1.28 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dj.1.29 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-120.b.1.13 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.b.1.14 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.b.2.15 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.b.2.16 | $120$ | $2$ | $2$ | $2$ |