Invariants
| Level: | $312$ | $\SL_2$-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 | $6^{12}$ | 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: | 6F1 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}72&5\\95&276\end{bmatrix}$, $\begin{bmatrix}113&90\\294&125\end{bmatrix}$, $\begin{bmatrix}279&128\\134&141\end{bmatrix}$, $\begin{bmatrix}281&12\\294&269\end{bmatrix}$, $\begin{bmatrix}282&161\\179&246\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.72.1.l.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: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 27 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^3}\cdot\frac{(y^{2}+81z^{2})^{3}(y^{6}+6075y^{4}z^{2}-295245y^{2}z^{4}+4782969z^{6})^{3}}{z^{2}y^{6}(y^{2}-243z^{2})^{6}(y^{2}-27z^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 312.48.0-12.h.1.5 | $312$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 312.48.0-12.h.1.6 | $312$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 312.72.0-6.a.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.72.0-6.a.1.7 | $312$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 312.288.5-12.q.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.u.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.y.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-12.bc.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.ei.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.ek.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.fs.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.fw.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-156.fy.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.gu.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.ht.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.iv.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bhz.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bin.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bss.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.btg.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |