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: | $\begin{bmatrix}43&18\\28&59\end{bmatrix}$, $\begin{bmatrix}55&76\\92&9\end{bmatrix}$, $\begin{bmatrix}97&80\\60&1\end{bmatrix}$, $\begin{bmatrix}103&60\\12&7\end{bmatrix}$, $\begin{bmatrix}109&68\\108&97\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.j.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 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^2}\cdot\frac{(3x+y)^{48}(81x^{8}-648x^{7}y+3024x^{6}y^{2}-6048x^{5}y^{3}+10080x^{4}y^{4}-8064x^{3}y^{5}+5376x^{2}y^{6}-1536xy^{7}+256y^{8})^{3}(81x^{8}+648x^{7}y+3024x^{6}y^{2}+6048x^{5}y^{3}+10080x^{4}y^{4}+8064x^{3}y^{5}+5376x^{2}y^{6}+1536xy^{7}+256y^{8})^{3}}{y^{4}x^{4}(3x+y)^{48}(3x^{2}-4y^{2})^{8}(3x^{2}+4y^{2})^{4}(9x^{4}+72x^{2}y^{2}+16y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.e.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.e.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-12.c.1.13 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-12.c.1.15 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.i.1.10 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.i.1.21 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.t.2.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.y.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.be.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bg.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bw.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.by.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.cc.2.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.cd.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-24.bf.2.21 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-24.y.1.23 | $120$ | $4$ | $4$ | $7$ |
| 120.192.1-120.gc.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ge.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gs.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gu.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.io.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.iq.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.je.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jg.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.480.16-120.t.1.16 | $120$ | $5$ | $5$ | $16$ |