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: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8N0 |
Level structure
| $\GL_2(\Z/104\Z)$-generators: | $\begin{bmatrix}3&84\\60&25\end{bmatrix}$, $\begin{bmatrix}29&78\\40&59\end{bmatrix}$, $\begin{bmatrix}59&34\\4&13\end{bmatrix}$, $\begin{bmatrix}81&8\\44&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 104.48.0.l.2 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 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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 104.48.0-52.c.1.11 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-52.c.1.16 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-8.e.1.4 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.i.2.12 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.i.2.24 | $104$ | $2$ | $2$ | $0$ | $?$ |
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.t.2.3 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.y.1.3 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.be.1.5 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bg.2.2 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.bw.1.6 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.by.2.4 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.cc.2.6 | $104$ | $2$ | $2$ | $1$ |
| 104.192.1-104.cd.1.4 | $104$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gb.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gf.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hg.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hk.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mi.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mm.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.no.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ns.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.cl.2.39 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.cm.2.42 | $312$ | $4$ | $4$ | $7$ |