Invariants
Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12L3 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}29&100\\140&9\end{bmatrix}$, $\begin{bmatrix}149&204\\62&103\end{bmatrix}$, $\begin{bmatrix}211&222\\294&13\end{bmatrix}$, $\begin{bmatrix}223&122\\66&173\end{bmatrix}$, $\begin{bmatrix}247&302\\24&233\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.96.3.fn.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $56$ |
Cyclic 312-torsion field degree: | $2688$ |
Full 312-torsion field degree: | $10063872$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
3.8.0-3.a.1.1 | $3$ | $24$ | $24$ | $0$ | $0$ |
104.24.0-104.b.1.5 | $104$ | $8$ | $8$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.96.0-12.a.2.15 | $12$ | $2$ | $2$ | $0$ | $0$ |
312.96.0-12.a.2.15 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.96.1-312.dh.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.96.1-312.dh.1.16 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.96.2-312.b.2.2 | $312$ | $2$ | $2$ | $2$ | $?$ |
312.96.2-312.b.2.16 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.384.5-312.is.2.2 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.iu.3.4 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.iu.4.7 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ix.3.3 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ix.4.6 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.iz.3.4 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.iz.4.2 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kc.1.4 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kc.3.5 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kf.1.3 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kf.4.6 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kq.1.4 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kq.3.7 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kt.1.6 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kt.3.4 | $312$ | $2$ | $2$ | $5$ |