Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| 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: | $\begin{bmatrix}7&108\\106&107\end{bmatrix}$, $\begin{bmatrix}17&42\\3&101\end{bmatrix}$, $\begin{bmatrix}29&24\\108&89\end{bmatrix}$, $\begin{bmatrix}55&24\\43&47\end{bmatrix}$, $\begin{bmatrix}107&114\\78&113\end{bmatrix}$, $\begin{bmatrix}113&72\\71&85\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.72.1.d.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: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 1 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{2}+3z^{2})^{3}(y^{6}+9y^{4}z^{2}+243y^{2}z^{4}+243z^{6})^{3}}{z^{4}y^{12}(y^{2}+z^{2})(y^{2}+9z^{2})^{3}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.48.0-12.f.1.5 | $120$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 120.48.0-12.f.1.7 | $120$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 120.72.0-6.a.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.288.3-24.d.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.d.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.e.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.e.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.h.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.h.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.i.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.i.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-12.a.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.i.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.q.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.r.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.y.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.ch.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ds.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.dt.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ei.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ej.1.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.fo.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.fv.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.zw.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bad.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bfk.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bfr.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-24.xo.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-24.xo.1.10 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-24.xp.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-24.xp.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dkh.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dkh.1.18 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dki.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dki.1.32 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |