Invariants
| Level: | $312$ | $\SL_2$-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
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}37&216\\90&5\end{bmatrix}$, $\begin{bmatrix}53&180\\198&269\end{bmatrix}$, $\begin{bmatrix}119&216\\198&157\end{bmatrix}$, $\begin{bmatrix}161&48\\192&151\end{bmatrix}$, $\begin{bmatrix}193&174\\270&161\end{bmatrix}$, $\begin{bmatrix}209&264\\108&311\end{bmatrix}$, $\begin{bmatrix}293&54\\60&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.72.1.a.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $13418496$ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 312.48.0-6.a.1.4 | $312$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 312.48.0-6.a.1.9 | $312$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 312.72.0-6.a.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.72.0-6.a.1.8 | $312$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 312.288.3-12.a.1.4 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-12.a.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-12.a.1.8 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-12.a.1.10 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-12.a.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-12.a.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-24.a.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-24.a.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-24.a.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-24.a.1.24 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-24.a.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-24.a.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-156.a.1.7 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-156.a.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-156.a.1.20 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-156.a.1.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-156.a.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-156.a.1.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.a.1.7 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.a.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.a.1.36 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.a.1.46 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.a.1.56 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.a.1.58 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.5-12.a.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.a.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.a.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.a.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.a.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.a.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.a.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.a.1.25 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.b.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.b.1.32 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.b.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.b.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.d.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.d.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.d.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.d.1.21 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.e.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.e.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.e.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.e.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.f.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.f.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.f.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.f.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.m.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.m.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.m.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.m.1.23 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.p.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.p.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.p.1.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.p.1.25 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.7-12.o.1.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-12.o.1.5 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-156.o.1.7 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-156.o.1.11 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.co.1.4 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.co.1.15 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-24.cw.1.5 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-24.cw.1.7 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |