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: | $\begin{bmatrix}5&58\\88&67\end{bmatrix}$, $\begin{bmatrix}11&36\\98&77\end{bmatrix}$, $\begin{bmatrix}52&59\\31&56\end{bmatrix}$, $\begin{bmatrix}59&66\\76&81\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 104.12.0.z.1 for the level structure with $-I$) |
| Cyclic 104-isogeny field degree: | $28$ |
| Cyclic 104-torsion field degree: | $1344$ |
| 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.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 104.12.0-4.c.1.6 | $104$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 104.48.0-104.m.1.9 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.o.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.w.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.x.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bn.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bo.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bs.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bv.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-104.bz.1.28 | $104$ | $14$ | $14$ | $11$ |
| 312.48.0-312.bv.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bx.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cd.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cf.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dl.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dm.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ds.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dv.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.72.2-312.cv.1.50 | $312$ | $3$ | $3$ | $2$ |
| 312.96.1-312.zv.1.60 | $312$ | $4$ | $4$ | $1$ |