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Properties

Label 204.48.0.c.3
Level $204$
Index $48$
Genus $0$
Cusps $10$
$\Q$-cusps $2$

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Invariants

Level: $204$ $\SL_2$-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

$\GL_2(\Z/204\Z)$-generators: $\begin{bmatrix}30&71\\73&152\end{bmatrix}$, $\begin{bmatrix}89&42\\134&157\end{bmatrix}$, $\begin{bmatrix}121&74\\154&177\end{bmatrix}$, $\begin{bmatrix}138&149\\61&98\end{bmatrix}$, $\begin{bmatrix}169&176\\64&69\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 204.96.0-204.c.3.1, 204.96.0-204.c.3.2, 204.96.0-204.c.3.3, 204.96.0-204.c.3.4, 204.96.0-204.c.3.5, 204.96.0-204.c.3.6, 204.96.0-204.c.3.7, 204.96.0-204.c.3.8, 204.96.0-204.c.3.9, 204.96.0-204.c.3.10, 204.96.0-204.c.3.11, 204.96.0-204.c.3.12, 204.96.0-204.c.3.13, 204.96.0-204.c.3.14, 204.96.0-204.c.3.15, 204.96.0-204.c.3.16
Cyclic 204-isogeny field degree: $18$
Cyclic 204-torsion field degree: $1152$
Full 204-torsion field degree: $7520256$

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
204.96.1.b.4 $204$ $2$ $2$ $1$
204.96.1.i.2 $204$ $2$ $2$ $1$
204.96.1.j.2 $204$ $2$ $2$ $1$
204.96.1.k.1 $204$ $2$ $2$ $1$
204.96.1.l.3 $204$ $2$ $2$ $1$
204.96.1.m.1 $204$ $2$ $2$ $1$
204.96.1.n.4 $204$ $2$ $2$ $1$
204.96.1.o.3 $204$ $2$ $2$ $1$
204.144.3.c.2 $204$ $3$ $3$ $3$