Invariants
Level: | $228$ | $\SL_2$-level: | $6$ | ||||
Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 6C0 |
Level structure
$\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}105&154\\179&151\end{bmatrix}$, $\begin{bmatrix}147&20\\125&81\end{bmatrix}$, $\begin{bmatrix}204&221\\215&213\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 228.8.0.b.1 for the level structure with $-I$) |
Cyclic 228-isogeny field degree: | $120$ |
Cyclic 228-torsion field degree: | $8640$ |
Full 228-torsion field degree: | $35458560$ |
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 |
---|---|---|---|---|---|
12.8.0-3.a.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
57.8.0-3.a.1.2 | $57$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
228.48.0-228.p.1.11 | $228$ | $3$ | $3$ | $0$ |
228.48.1-228.g.1.3 | $228$ | $3$ | $3$ | $1$ |
228.64.1-228.d.1.5 | $228$ | $4$ | $4$ | $1$ |
228.320.11-228.f.1.16 | $228$ | $20$ | $20$ | $11$ |