Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8N0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}3&50\\80&61\end{bmatrix}$, $\begin{bmatrix}11&94\\12&61\end{bmatrix}$, $\begin{bmatrix}63&100\\116&39\end{bmatrix}$, $\begin{bmatrix}117&56\\116&81\end{bmatrix}$, $\begin{bmatrix}119&8\\92&97\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.n.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $368640$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^2}\cdot\frac{x^{48}(25458241x^{16}+230810336x^{15}y+1017129120x^{14}y^{2}+2743807360x^{13}y^{3}+5152387520x^{12}y^{4}+7191141888x^{11}y^{5}+7662151168x^{10}y^{6}+6317004800x^{9}y^{7}+4112745984x^{8}y^{8}+2129764352x^{7}y^{9}+881090560x^{6}y^{10}+267681792x^{5}y^{11}+62504960x^{4}y^{12}+10092544x^{3}y^{13}+2752512x^{2}y^{14}+524288xy^{15}+65536y^{16})^{3}}{x^{52}(x+2y)^{4}(5x^{2}-4xy-4y^{2})^{8}(7x^{2}+4xy+4y^{2})^{4}(73x^{4}+152x^{3}y+168x^{2}y^{2}+32xy^{3}+16y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.e.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.e.1.6 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.e.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.e.1.18 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.i.2.8 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.i.2.32 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.192.1-24.u.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.z.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bm.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bo.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bx.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bz.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.cg.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.ch.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-24.bn.2.10 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-24.bd.2.2 | $120$ | $4$ | $4$ | $7$ |
| 120.192.1-120.hj.2.13 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hl.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hz.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ib.2.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jv.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jx.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.kl.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.kn.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.480.16-120.bb.2.30 | $120$ | $5$ | $5$ | $16$ |