Invariants
| Level: | $104$ | $\SL_2$-level: | $8$ | ||||
| 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 | $1^{2}\cdot2\cdot8$ | 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: | 8C0 |
Level structure
| $\GL_2(\Z/104\Z)$-generators: | $\begin{bmatrix}44&33\\15&94\end{bmatrix}$, $\begin{bmatrix}44&61\\33&12\end{bmatrix}$, $\begin{bmatrix}48&77\\53&84\end{bmatrix}$, $\begin{bmatrix}49&90\\32&91\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 104.12.0.ba.1 for the level structure with $-I$) |
| Cyclic 104-isogeny field degree: | $28$ |
| Cyclic 104-torsion field degree: | $672$ |
| 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 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 |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 52.12.0-4.c.1.2 | $52$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 104.48.0-104.m.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.n.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.be.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bg.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bj.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bk.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bw.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bz.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-104.ce.1.3 | $104$ | $14$ | $14$ | $11$ |
| 312.48.0-312.by.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ca.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cg.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ci.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dp.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dq.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ea.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ed.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.72.2-312.di.1.23 | $312$ | $3$ | $3$ | $2$ |
| 312.96.1-312.baa.1.41 | $312$ | $4$ | $4$ | $1$ |