Invariants
Level: | $228$ | $\SL_2$-level: | $12$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E0 |
Level structure
$\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}112&17\\61&66\end{bmatrix}$, $\begin{bmatrix}135&218\\164&45\end{bmatrix}$, $\begin{bmatrix}141&166\\14&175\end{bmatrix}$, $\begin{bmatrix}143&90\\180&59\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 228.24.0.q.1 for the level structure with $-I$) |
Cyclic 228-isogeny field degree: | $40$ |
Cyclic 228-torsion field degree: | $1440$ |
Full 228-torsion field degree: | $11819520$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
228.24.0-6.a.1.3 | $228$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
228.96.1-228.c.1.19 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.f.1.10 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.z.1.6 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.ba.1.5 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.bh.1.1 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.bi.1.3 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.bt.1.7 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.bu.1.10 | $228$ | $2$ | $2$ | $1$ |
228.144.1-228.r.1.4 | $228$ | $3$ | $3$ | $1$ |