Invariants
| Level: | $120$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6D1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}9&119\\35&24\end{bmatrix}$, $\begin{bmatrix}11&9\\96&71\end{bmatrix}$, $\begin{bmatrix}27&50\\14&87\end{bmatrix}$, $\begin{bmatrix}63&83\\32&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.bx.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $72$ |
| Cyclic 120-torsion field degree: | $2304$ |
| Full 120-torsion field degree: | $737280$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 216 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{(y^{2}-1944z^{2})^{3}(y^{2}-216z^{2})}{z^{2}y^{6}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.2.0.a.1 | $8$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 15.24.0-3.a.1.1 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.24.0-3.a.1.1 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 120.16.0-24.a.1.1 | $120$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 120.16.0-24.a.1.5 | $120$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 120.24.0-3.a.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.144.1-24.c.1.2 | $120$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 120.192.5-24.dl.1.2 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 120.240.9-120.dt.1.5 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-120.ipd.1.3 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-120.brr.1.7 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |