Invariants
Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E5 |
Level structure
$\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}23&6\\176&115\end{bmatrix}$, $\begin{bmatrix}117&22\\128&49\end{bmatrix}$, $\begin{bmatrix}145&194\\150&47\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 228.192.5.q.2 for the level structure with $-I$) |
Cyclic 228-isogeny field degree: | $40$ |
Cyclic 228-torsion field degree: | $1440$ |
Full 228-torsion field degree: | $1477440$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.192.3-12.f.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ |
228.192.1-228.a.2.2 | $228$ | $2$ | $2$ | $1$ | $?$ |
228.192.1-228.a.2.14 | $228$ | $2$ | $2$ | $1$ | $?$ |
228.192.1-228.f.2.1 | $228$ | $2$ | $2$ | $1$ | $?$ |
228.192.1-228.f.2.16 | $228$ | $2$ | $2$ | $1$ | $?$ |
228.192.1-228.f.3.2 | $228$ | $2$ | $2$ | $1$ | $?$ |
228.192.1-228.f.3.14 | $228$ | $2$ | $2$ | $1$ | $?$ |
228.192.3-228.b.1.3 | $228$ | $2$ | $2$ | $3$ | $?$ |
228.192.3-228.b.1.7 | $228$ | $2$ | $2$ | $3$ | $?$ |
228.192.3-12.f.1.4 | $228$ | $2$ | $2$ | $3$ | $?$ |
228.192.3-228.t.2.4 | $228$ | $2$ | $2$ | $3$ | $?$ |
228.192.3-228.t.2.11 | $228$ | $2$ | $2$ | $3$ | $?$ |
228.192.3-228.t.2.13 | $228$ | $2$ | $2$ | $3$ | $?$ |