Invariants
Level: | $204$ | $\SL_2$-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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
$\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}27&178\\100&27\end{bmatrix}$, $\begin{bmatrix}48&197\\199&122\end{bmatrix}$, $\begin{bmatrix}111&94\\200&175\end{bmatrix}$, $\begin{bmatrix}182&159\\135&140\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 102.24.0.b.1 for the level structure with $-I$) |
Cyclic 204-isogeny field degree: | $36$ |
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 |
---|---|---|---|---|---|
12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
204.24.0-6.a.1.7 | $204$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
204.96.1-204.m.1.5 | $204$ | $2$ | $2$ | $1$ |
204.96.1-204.o.1.12 | $204$ | $2$ | $2$ | $1$ |
204.96.1-204.bd.1.7 | $204$ | $2$ | $2$ | $1$ |
204.96.1-204.be.1.7 | $204$ | $2$ | $2$ | $1$ |
204.96.1-204.bl.1.5 | $204$ | $2$ | $2$ | $1$ |
204.96.1-204.bm.1.5 | $204$ | $2$ | $2$ | $1$ |
204.96.1-204.bs.1.7 | $204$ | $2$ | $2$ | $1$ |
204.96.1-204.bu.1.5 | $204$ | $2$ | $2$ | $1$ |
204.144.1-102.c.1.3 | $204$ | $3$ | $3$ | $1$ |