Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $36$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $3^{2}\cdot12$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}13&58\\10&71\end{bmatrix}$, $\begin{bmatrix}30&17\\89&80\end{bmatrix}$, $\begin{bmatrix}31&94\\70&17\end{bmatrix}$, $\begin{bmatrix}73&40\\108&83\end{bmatrix}$, $\begin{bmatrix}119&36\\96&65\end{bmatrix}$, $\begin{bmatrix}119&106\\106&57\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.18.1.b.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $983040$ |
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 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(4y^{2}+3z^{2})^{3}}{z^{4}(y^{2}+z^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.12.0-4.b.1.3 | $40$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.12.0-4.b.1.3 | $40$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 120.72.1-12.g.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-12.h.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-12.k.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-12.l.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.2-12.a.1.21 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.i.1.11 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.q.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.r.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.u.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.v.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.y.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.z.1.4 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.1-24.y.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bb.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bk.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bn.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.2-24.i.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.z.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.bw.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.bz.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cy.1.4 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.db.1.4 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dk.1.2 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dn.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.1-60.g.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-60.h.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-60.k.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-60.l.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.2-60.q.1.27 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.r.1.15 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.u.1.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.v.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.y.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.z.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.bc.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.bd.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.180.7-60.b.1.5 | $120$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 120.216.7-60.b.1.27 | $120$ | $6$ | $6$ | $7$ | $?$ | not computed |
| 120.360.13-60.bd.1.29 | $120$ | $10$ | $10$ | $13$ | $?$ | not computed |
| 120.72.1-120.y.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bb.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bk.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bn.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.2-120.bw.1.15 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.bz.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.ci.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cl.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dk.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dn.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dw.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dz.1.3 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |