Invariants
| Level: | $312$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}13&18\\42&305\end{bmatrix}$, $\begin{bmatrix}76&231\\257&254\end{bmatrix}$, $\begin{bmatrix}139&14\\282&11\end{bmatrix}$, $\begin{bmatrix}203&280\\226&225\end{bmatrix}$, $\begin{bmatrix}263&170\\58&135\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.12.0.ba.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $80510976$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1542 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot3}\cdot\frac{(3x+y)^{12}(9x^{4}-192x^{2}y^{2}+256y^{4})^{3}}{y^{8}x^{2}(3x+y)^{12}(3x^{2}-64y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 52.12.0-4.c.1.1 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 312.12.0-4.c.1.3 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.48.0-24.m.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.n.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bc.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.be.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bh.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bi.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bs.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bv.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cu.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cw.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cy.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.da.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ds.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.du.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ea.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ec.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.72.2-24.cu.1.8 | $312$ | $3$ | $3$ | $2$ |
| 312.96.1-24.iu.1.21 | $312$ | $4$ | $4$ | $1$ |
| 312.336.11-312.cs.1.53 | $312$ | $14$ | $14$ | $11$ |