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Properties

Label 228.48.0.c.2
Level $228$
Index $48$
Genus $0$
Cusps $10$
$\Q$-cusps $2$

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Invariants

Level: $228$ $\SL_2$-level: $12$
Index: $48$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $2$ are rational) Cusp widths $1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$ Cusp orbits $1^{2}\cdot2^{4}$
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: 12J0

Level structure

$\GL_2(\Z/228\Z)$-generators: $\begin{bmatrix}39&34\\70&219\end{bmatrix}$, $\begin{bmatrix}68&135\\45&62\end{bmatrix}$, $\begin{bmatrix}83&84\\90&161\end{bmatrix}$, $\begin{bmatrix}154&173\\159&152\end{bmatrix}$, $\begin{bmatrix}159&164\\140&207\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 228.96.0-228.c.2.1, 228.96.0-228.c.2.2, 228.96.0-228.c.2.3, 228.96.0-228.c.2.4, 228.96.0-228.c.2.5, 228.96.0-228.c.2.6, 228.96.0-228.c.2.7, 228.96.0-228.c.2.8, 228.96.0-228.c.2.9, 228.96.0-228.c.2.10, 228.96.0-228.c.2.11, 228.96.0-228.c.2.12, 228.96.0-228.c.2.13, 228.96.0-228.c.2.14, 228.96.0-228.c.2.15, 228.96.0-228.c.2.16
Cyclic 228-isogeny field degree: $20$
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
$X_0(12)$ $12$ $2$ $2$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
228.96.1.b.4 $228$ $2$ $2$ $1$
228.96.1.i.2 $228$ $2$ $2$ $1$
228.96.1.j.2 $228$ $2$ $2$ $1$
228.96.1.k.1 $228$ $2$ $2$ $1$
228.96.1.l.2 $228$ $2$ $2$ $1$
228.96.1.m.3 $228$ $2$ $2$ $1$
228.96.1.n.1 $228$ $2$ $2$ $1$
228.96.1.o.3 $228$ $2$ $2$ $1$
228.144.3.c.2 $228$ $3$ $3$ $3$