Invariants
| Level: | $204$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $36$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $6^{3}$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C1 |
Level structure
| $\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}3&190\\100&127\end{bmatrix}$, $\begin{bmatrix}17&4\\16&195\end{bmatrix}$, $\begin{bmatrix}87&46\\110&9\end{bmatrix}$, $\begin{bmatrix}101&50\\194&49\end{bmatrix}$, $\begin{bmatrix}103&152\\134&129\end{bmatrix}$, $\begin{bmatrix}115&100\\28&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.18.1.a.1 for the level structure with $-I$) |
| Cyclic 204-isogeny field degree: | $144$ |
| Cyclic 204-torsion field degree: | $9216$ |
| Full 204-torsion field degree: | $10027008$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 1 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(y^{2}+3z^{2})^{3}}{z^{2}(y-z)^{2}(y+z)^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.12.0-2.a.1.2 | $68$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.12.0-2.a.1.2 | $68$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 204.12.1-6.a.1.4 | $204$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 204.72.1-6.a.1.3 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.1-6.b.1.6 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.1-12.a.1.3 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.1-12.b.1.3 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.2-12.a.1.8 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-12.b.1.12 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-12.c.1.8 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-12.d.1.8 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-12.e.1.7 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-12.f.1.7 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-12.g.1.8 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-12.h.1.8 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.1-102.a.1.5 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.1-102.b.1.5 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.1-204.a.1.5 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.1-204.b.1.5 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.2-204.a.1.5 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-204.b.1.2 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-204.c.1.2 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-204.d.1.4 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-204.e.1.8 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-204.f.1.8 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-204.g.1.5 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.72.2-204.h.1.7 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |