Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| 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 $6$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}11&12\\78&43\end{bmatrix}$, $\begin{bmatrix}23&18\\84&67\end{bmatrix}$, $\begin{bmatrix}23&24\\72&5\end{bmatrix}$, $\begin{bmatrix}31&96\\54&115\end{bmatrix}$, $\begin{bmatrix}71&48\\18&65\end{bmatrix}$, $\begin{bmatrix}83&108\\6&23\end{bmatrix}$, $\begin{bmatrix}109&108\\24&119\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.72.1.a.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: | 36.2.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{2}+3z^{2})^{3}(y^{6}+225y^{4}z^{2}-405y^{2}z^{4}+243z^{6})^{3}}{z^{2}y^{6}(y-3z)^{6}(y-z)^{2}(y+z)^{2}(y+3z)^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.12.0-2.a.1.1 | $40$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.48.0-6.a.1.7 | $120$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 120.72.1-6.b.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-6.b.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.288.3-12.a.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-12.a.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-12.a.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-12.a.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-12.a.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-12.a.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.a.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.a.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.a.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.a.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.a.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-24.a.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-60.a.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-60.a.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-60.a.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-60.a.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-60.a.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-60.a.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.a.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.a.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.a.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.a.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.a.1.42 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.a.1.59 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-12.a.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.a.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.a.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.a.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.a.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.a.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.a.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.a.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.a.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.a.1.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.a.1.23 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.b.1.22 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.b.1.27 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.b.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.b.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.b.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.b.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.d.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.d.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.d.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.d.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.d.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.d.1.25 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.e.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.e.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.e.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.e.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.e.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.e.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.f.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.f.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.f.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.f.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.f.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.f.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.m.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.m.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.m.1.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.m.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.m.1.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.m.1.19 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.p.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.p.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-24.p.1.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.p.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.p.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.p.1.17 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-12.o.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-12.o.1.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.o.1.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.o.1.7 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.co.1.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.co.1.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-24.cw.1.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-24.cw.1.11 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |