Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}41&59\\46&53\end{bmatrix}$, $\begin{bmatrix}43&49\\66&107\end{bmatrix}$, $\begin{bmatrix}47&73\\70&103\end{bmatrix}$, $\begin{bmatrix}71&94\\40&87\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.i.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $737280$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 y^{2} + z^{2} + w^{2} $ |
| $=$ | $4 x^{2} - 3 y z$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} + y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(3z^{2}-w^{2})^{3}}{z^{2}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.24.1.i.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{3}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}+Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-4.c.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.24.0-4.c.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.144.5-24.bi.1.2 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.192.5-24.w.1.7 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 120.240.9-120.q.1.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-120.cka.1.11 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-120.oh.1.3 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |