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: | $\begin{bmatrix}31&56\\30&91\end{bmatrix}$, $\begin{bmatrix}33&40\\46&35\end{bmatrix}$, $\begin{bmatrix}41&64\\86&95\end{bmatrix}$, $\begin{bmatrix}71&20\\46&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 104.48.0.x.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.1.16 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.i.2.20 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.i.2.27 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.m.1.18 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.m.1.19 | $104$ | $2$ | $2$ | $0$ | $?$ |
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.s.2.2 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.t.2.2 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.x.1.5 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.y.1.2 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bm.1.5 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bn.1.6 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bo.2.3 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bp.2.4 | $104$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jc.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jd.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jg.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jh.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ki.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.kj.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.km.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.kn.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.of.2.48 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.ib.2.38 | $312$ | $4$ | $4$ | $7$ |