Invariants
| Level: | $312$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8J0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}81&52\\262&231\end{bmatrix}$, $\begin{bmatrix}97&188\\84&55\end{bmatrix}$, $\begin{bmatrix}111&272\\82&193\end{bmatrix}$, $\begin{bmatrix}119&216\\28&253\end{bmatrix}$, $\begin{bmatrix}175&284\\90&179\end{bmatrix}$, $\begin{bmatrix}201&152\\202&255\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $40255488$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 220 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}-16x^{6}y^{2}+320x^{4}y^{4}-2048x^{2}y^{6}+4096y^{8})^{3}}{y^{4}x^{32}(x-2y)^{2}(x+2y)^{2}(x^{2}-8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 312.24.0-4.b.1.1 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-4.b.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 |
|---|---|---|---|---|
| 312.96.0-8.b.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-8.c.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-8.e.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-8.f.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-8.h.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-8.i.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.i.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.j.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-8.k.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.k.2.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-8.l.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.l.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.m.2.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.n.2.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.o.2.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.p.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.r.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.s.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.s.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.t.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.v.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.w.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.w.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.x.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.be.2.19 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bg.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bm.2.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bo.2.21 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bu.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bw.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cc.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ce.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-8.i.2.6 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-8.k.2.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-8.m.2.6 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-8.n.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.be.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-104.be.2.13 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bf.2.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-104.bf.2.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bi.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-104.bi.2.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bj.2.12 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-104.bj.2.13 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dx.2.27 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dz.2.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.ef.2.18 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.eh.2.27 | $312$ | $2$ | $2$ | $1$ |
| 312.144.4-24.z.2.14 | $312$ | $3$ | $3$ | $4$ |
| 312.192.3-24.bq.2.36 | $312$ | $4$ | $4$ | $3$ |