Invariants
Level: | $228$ | $\SL_2$-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 $4$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
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: | 12L3 |
Level structure
$\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}11&142\\116&123\end{bmatrix}$, $\begin{bmatrix}89&136\\92&51\end{bmatrix}$, $\begin{bmatrix}203&112\\56&117\end{bmatrix}$, $\begin{bmatrix}223&92\\12&113\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 228.96.3.l.1 for the level structure with $-I$) |
Cyclic 228-isogeny field degree: | $20$ |
Cyclic 228-torsion field degree: | $1440$ |
Full 228-torsion field degree: | $2954880$ |
Rational points
This modular curve has 4 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 | $3$ | $24$ | $24$ | $0$ | $0$ |
76.24.0-4.b.1.2 | $76$ | $8$ | $8$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.96.1-12.b.1.12 | $12$ | $2$ | $2$ | $1$ | $0$ |
228.96.0-228.a.1.6 | $228$ | $2$ | $2$ | $0$ | $?$ |
228.96.0-228.a.1.8 | $228$ | $2$ | $2$ | $0$ | $?$ |
228.96.1-12.b.1.7 | $228$ | $2$ | $2$ | $1$ | $?$ |
228.96.2-228.a.1.7 | $228$ | $2$ | $2$ | $2$ | $?$ |
228.96.2-228.a.1.8 | $228$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
228.384.5-228.b.2.4 | $228$ | $2$ | $2$ | $5$ |
228.384.5-228.e.1.2 | $228$ | $2$ | $2$ | $5$ |
228.384.5-228.e.4.4 | $228$ | $2$ | $2$ | $5$ |
228.384.5-228.h.1.2 | $228$ | $2$ | $2$ | $5$ |
228.384.5-228.h.4.1 | $228$ | $2$ | $2$ | $5$ |
228.384.5-228.k.1.4 | $228$ | $2$ | $2$ | $5$ |
228.384.5-228.k.4.3 | $228$ | $2$ | $2$ | $5$ |