Invariants
| Level: | $104$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/104\Z)$-generators: | $\begin{bmatrix}15&32\\2&49\end{bmatrix}$, $\begin{bmatrix}29&60\\94&7\end{bmatrix}$, $\begin{bmatrix}39&32\\46&99\end{bmatrix}$, $\begin{bmatrix}45&52\\58&99\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 104.48.0.w.1 for the level structure with $-I$) |
| Cyclic 104-isogeny field degree: | $28$ |
| Cyclic 104-torsion field degree: | $1344$ |
| Full 104-torsion field degree: | $419328$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 104.48.0-8.e.1.8 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.h.1.22 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.h.1.27 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.m.1.1 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.m.1.18 | $104$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 104.192.1-104.h.2.2 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.i.1.8 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.x.1.8 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.y.1.2 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bk.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bl.1.8 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bo.1.7 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bp.1.2 | $104$ | $2$ | $2$ | $1$ |
| 312.192.1-312.iy.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.iz.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.je.2.5 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jf.2.13 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ke.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.kf.2.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.kk.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.kl.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.ob.2.29 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.hz.2.55 | $312$ | $4$ | $4$ | $7$ |