Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $3^{8}\cdot12^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| 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: | 12S1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}25&66\\4&35\end{bmatrix}$, $\begin{bmatrix}41&78\\110&97\end{bmatrix}$, $\begin{bmatrix}65&114\\18&23\end{bmatrix}$, $\begin{bmatrix}67&78\\64&119\end{bmatrix}$, $\begin{bmatrix}107&6\\83&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.1.n.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $245760$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.e |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 8 $ |
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 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{(y^{2}-24z^{2})^{3}(y^{6}-72y^{4}z^{2}+15552y^{2}z^{4}-124416z^{6})^{3}}{z^{4}y^{12}(y^{2}-72z^{2})^{3}(y^{2}-8z^{2})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.6.0.c.1 | $8$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 15.24.0-3.a.1.1 | $15$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.72.0-6.a.1.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 120.48.0-24.bw.1.1 | $120$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 120.48.0-24.bw.1.5 | $120$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 120.72.0-6.a.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.288.5-24.a.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.bu.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.cf.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.cj.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.fr.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.fs.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.gb.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.gd.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.zz.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.baa.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bag.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bah.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bfn.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bfo.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bfu.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bfv.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |