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: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
Level structure
| $\GL_2(\Z/104\Z)$-generators: | $\begin{bmatrix}13&18\\34&29\end{bmatrix}$, $\begin{bmatrix}22&55\\35&34\end{bmatrix}$, $\begin{bmatrix}35&58\\94&79\end{bmatrix}$, $\begin{bmatrix}58&71\\63&82\end{bmatrix}$, $\begin{bmatrix}87&6\\64&85\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 104-isogeny field degree: | $14$ |
| 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, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 104.12.0-4.c.1.1 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 104.12.0-4.c.1.4 | $104$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 104.48.0-8.i.1.12 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-8.k.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-8.q.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-8.r.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-8.ba.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-8.ba.2.7 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-8.bb.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-8.bb.2.8 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bj.1.10 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bl.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bn.1.10 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bp.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.ca.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.ca.2.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.cb.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.cb.2.3 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-104.bx.1.21 | $104$ | $14$ | $14$ | $11$ |
| 208.48.0-16.e.1.12 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.e.2.3 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.f.1.10 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.f.2.6 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.g.1.12 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.h.1.12 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.m.1.3 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.m.2.13 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.n.1.5 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.n.2.7 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.o.1.17 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.p.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.48.1-16.a.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-208.a.1.32 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-16.b.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-208.b.1.16 | $208$ | $2$ | $2$ | $1$ |
| 312.48.0-24.bh.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bj.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bl.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bn.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.by.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.by.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bz.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bz.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dd.1.19 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.df.1.18 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dh.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dj.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ei.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ei.2.18 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ej.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ej.2.18 | $312$ | $2$ | $2$ | $0$ |
| 312.72.2-24.cj.1.33 | $312$ | $3$ | $3$ | $2$ |
| 312.96.1-24.ir.1.1 | $312$ | $4$ | $4$ | $1$ |