Invariants
Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
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: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12D3 |
Level structure
$\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}41&165\\30&89\end{bmatrix}$, $\begin{bmatrix}55&63\\210&193\end{bmatrix}$, $\begin{bmatrix}91&169\\144&65\end{bmatrix}$, $\begin{bmatrix}101&27\\132&17\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 228.72.3.mb.1 for the level structure with $-I$) |
Cyclic 228-isogeny field degree: | $40$ |
Cyclic 228-torsion field degree: | $2880$ |
Full 228-torsion field degree: | $3939840$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.72.0-6.a.1.5 | $12$ | $2$ | $2$ | $0$ | $0$ |
114.72.0-6.a.1.1 | $114$ | $2$ | $2$ | $0$ | $?$ |
228.48.1-228.p.1.5 | $228$ | $3$ | $3$ | $1$ | $?$ |
228.48.1-228.p.1.11 | $228$ | $3$ | $3$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
228.288.5-228.a.1.5 | $228$ | $2$ | $2$ | $5$ |
228.288.5-228.bg.1.2 | $228$ | $2$ | $2$ | $5$ |
228.288.5-228.bw.1.3 | $228$ | $2$ | $2$ | $5$ |
228.288.5-228.cb.1.4 | $228$ | $2$ | $2$ | $5$ |
228.288.5-228.fp.1.4 | $228$ | $2$ | $2$ | $5$ |
228.288.5-228.fs.1.4 | $228$ | $2$ | $2$ | $5$ |
228.288.5-228.fy.1.4 | $228$ | $2$ | $2$ | $5$ |
228.288.5-228.ge.1.4 | $228$ | $2$ | $2$ | $5$ |