Invariants
Level: | $228$ | $\SL_2$-level: | $76$ | Newform level: | $1$ | ||
Index: | $240$ | $\PSL_2$-index: | $240$ | ||||
Genus: | $17 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot38^{2}\cdot76^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $4 \le \gamma \le 17$ | ||||||
$\overline{\Q}$-gonality: | $4 \le \gamma \le 17$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 76A17 |
Level structure
$\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}7&0\\82&173\end{bmatrix}$, $\begin{bmatrix}69&152\\67&199\end{bmatrix}$, $\begin{bmatrix}143&76\\58&189\end{bmatrix}$, $\begin{bmatrix}167&0\\39&187\end{bmatrix}$, $\begin{bmatrix}199&76\\126&115\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 228.480.17-228.l.1.1, 228.480.17-228.l.1.2, 228.480.17-228.l.1.3, 228.480.17-228.l.1.4, 228.480.17-228.l.1.5, 228.480.17-228.l.1.6, 228.480.17-228.l.1.7, 228.480.17-228.l.1.8, 228.480.17-228.l.1.9, 228.480.17-228.l.1.10, 228.480.17-228.l.1.11, 228.480.17-228.l.1.12 |
Cyclic 228-isogeny field degree: | $4$ |
Cyclic 228-torsion field degree: | $288$ |
Full 228-torsion field degree: | $2363904$ |
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 |
---|---|---|---|---|---|
12.12.0.h.1 | $12$ | $20$ | $20$ | $0$ | $0$ |
$X_0(19)$ | $19$ | $12$ | $12$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.12.0.h.1 | $12$ | $20$ | $20$ | $0$ | $0$ |
$X_0(76)$ | $76$ | $2$ | $2$ | $8$ | $?$ |
228.120.8.a.1 | $228$ | $2$ | $2$ | $8$ | $?$ |
228.120.9.f.1 | $228$ | $2$ | $2$ | $9$ | $?$ |