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Properties

Label 204.24.0.p.1
Level $204$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $204$ $\SL_2$-level: $12$
Index: $24$ $\PSL_2$-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

$\GL_2(\Z/204\Z)$-generators: $\begin{bmatrix}46&131\\199&108\end{bmatrix}$, $\begin{bmatrix}68&31\\65&78\end{bmatrix}$, $\begin{bmatrix}117&32\\10&83\end{bmatrix}$, $\begin{bmatrix}121&0\\46&95\end{bmatrix}$, $\begin{bmatrix}148&47\\115&156\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 204.48.0-204.p.1.1, 204.48.0-204.p.1.2, 204.48.0-204.p.1.3, 204.48.0-204.p.1.4, 204.48.0-204.p.1.5, 204.48.0-204.p.1.6, 204.48.0-204.p.1.7, 204.48.0-204.p.1.8, 204.48.0-204.p.1.9, 204.48.0-204.p.1.10, 204.48.0-204.p.1.11, 204.48.0-204.p.1.12, 204.48.0-204.p.1.13, 204.48.0-204.p.1.14, 204.48.0-204.p.1.15, 204.48.0-204.p.1.16
Cyclic 204-isogeny field degree: $36$
Cyclic 204-torsion field degree: $2304$
Full 204-torsion field degree: $15040512$

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

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
$X_0(3)$ $3$ $6$ $6$ $0$ $0$
68.6.0.b.1 $68$ $4$ $4$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(6)$ $6$ $2$ $2$ $0$ $0$
68.6.0.b.1 $68$ $4$ $4$ $0$ $0$

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

Covering curve Level Index Degree Genus
204.48.1.a.1 $204$ $2$ $2$ $1$
204.48.1.e.1 $204$ $2$ $2$ $1$
204.48.1.q.1 $204$ $2$ $2$ $1$
204.48.1.s.1 $204$ $2$ $2$ $1$
204.48.1.bk.1 $204$ $2$ $2$ $1$
204.48.1.bm.1 $204$ $2$ $2$ $1$
204.48.1.bp.1 $204$ $2$ $2$ $1$
204.48.1.bq.1 $204$ $2$ $2$ $1$
204.72.1.k.1 $204$ $3$ $3$ $1$