Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}72&5\\209&156\end{bmatrix}$, $\begin{bmatrix}84&107\\167&108\end{bmatrix}$, $\begin{bmatrix}152&91\\153&46\end{bmatrix}$, $\begin{bmatrix}185&136\\38&279\end{bmatrix}$, $\begin{bmatrix}246&245\\203&72\end{bmatrix}$, $\begin{bmatrix}255&32\\272&159\end{bmatrix}$, $\begin{bmatrix}276&85\\227&154\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $28$ |
| Cyclic 312-torsion field degree: | $2688$ |
| 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 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 312.12.0-4.c.1.2 | $312$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.96.0-12.c.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.c.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.c.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.c.2.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.c.3.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.c.3.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.c.4.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.c.4.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.1.37 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.2.14 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.2.33 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.3.20 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.3.27 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.4.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.c.4.38 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bs.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bs.1.20 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bs.2.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bs.2.23 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.1.20 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.2.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.2.23 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.2.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.3.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.3.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.4.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.4.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.1.17 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.1.48 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.2.56 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.1.17 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.1.48 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.2.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.2.60 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.1.17 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.1.46 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.2.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.2.58 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.3.30 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.3.33 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.4.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.4.54 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-12.b.1.17 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.h.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.k.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.k.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.l.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.l.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.o.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.p.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.cg.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.es.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ik.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.in.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iq.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iq.1.26 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ir.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ir.1.44 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.is.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.is.1.28 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.it.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.it.1.32 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iu.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iu.1.32 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iv.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iv.1.28 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iw.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iw.1.28 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ix.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ix.1.26 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zc.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zf.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zo.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zr.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zu.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zu.1.44 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zv.1.29 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zv.1.36 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zw.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zw.1.56 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zx.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zx.1.64 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zy.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zy.1.64 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zz.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zz.1.56 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.baa.1.29 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.baa.1.36 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bab.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bab.1.44 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-24.f.1.7 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.f.1.26 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.f.2.6 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.f.2.27 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.1.7 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.1.26 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.2.6 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.2.27 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.1.17 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.1.48 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.2.5 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.2.60 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.1.17 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.1.48 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.2.9 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.2.56 | $312$ | $2$ | $2$ | $2$ |
| 312.144.1-12.f.1.1 | $312$ | $3$ | $3$ | $1$ |