Invariants
| Level: | $104$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/104\Z)$-generators: | $\begin{bmatrix}49&44\\22&1\end{bmatrix}$, $\begin{bmatrix}75&80\\52&23\end{bmatrix}$, $\begin{bmatrix}77&100\\86&47\end{bmatrix}$, $\begin{bmatrix}97&54\\100&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 52.12.0.a.1 for the level structure with $-I$) |
| Cyclic 104-isogeny field degree: | $56$ |
| Cyclic 104-torsion field degree: | $2688$ |
| Full 104-torsion field degree: | $1677312$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 188 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^4\cdot13^2}\cdot\frac{x^{12}(169x^{4}+117x^{2}y^{2}+81y^{4})^{3}}{y^{4}x^{16}(13x^{2}+9y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 104.12.0-2.a.1.1 | $104$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 104.48.0-52.a.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-52.a.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.b.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.b.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-52.c.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-52.c.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.f.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.f.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-52.c.1.1 | $104$ | $14$ | $14$ | $11$ |
| 312.48.0-156.d.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.d.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.f.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.f.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.i.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.i.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.o.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.o.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.72.2-156.a.1.8 | $312$ | $3$ | $3$ | $2$ |
| 312.96.1-156.a.1.1 | $312$ | $4$ | $4$ | $1$ |