Invariants
Level: | $104$ | $\SL_2$-level: | $4$ | ||||
Index: | $12$ | $\PSL_2$-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 | $2^{2}\cdot4^{2}$ | 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: | 4E0 |
Level structure
$\GL_2(\Z/104\Z)$-generators: | $\begin{bmatrix}19&32\\36&7\end{bmatrix}$, $\begin{bmatrix}23&84\\48&7\end{bmatrix}$, $\begin{bmatrix}63&38\\100&101\end{bmatrix}$, $\begin{bmatrix}89&10\\86&61\end{bmatrix}$, $\begin{bmatrix}97&28\\72&35\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 104.24.0-104.b.1.1, 104.24.0-104.b.1.2, 104.24.0-104.b.1.3, 104.24.0-104.b.1.4, 104.24.0-104.b.1.5, 104.24.0-104.b.1.6, 104.24.0-104.b.1.7, 104.24.0-104.b.1.8, 312.24.0-104.b.1.1, 312.24.0-104.b.1.2, 312.24.0-104.b.1.3, 312.24.0-104.b.1.4, 312.24.0-104.b.1.5, 312.24.0-104.b.1.6, 312.24.0-104.b.1.7, 312.24.0-104.b.1.8 |
Cyclic 104-isogeny field degree: | $56$ |
Cyclic 104-torsion field degree: | $2688$ |
Full 104-torsion field degree: | $3354624$ |
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 |
---|---|---|---|---|---|
$X(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
104.6.0.d.1 | $104$ | $2$ | $2$ | $0$ | $?$ |
104.6.0.e.1 | $104$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
104.24.0.c.1 | $104$ | $2$ | $2$ | $0$ |
104.24.0.d.1 | $104$ | $2$ | $2$ | $0$ |
104.24.0.e.1 | $104$ | $2$ | $2$ | $0$ |
104.24.0.f.1 | $104$ | $2$ | $2$ | $0$ |
104.168.11.d.1 | $104$ | $14$ | $14$ | $11$ |
312.24.0.k.1 | $312$ | $2$ | $2$ | $0$ |
312.24.0.m.1 | $312$ | $2$ | $2$ | $0$ |
312.24.0.q.1 | $312$ | $2$ | $2$ | $0$ |
312.24.0.s.1 | $312$ | $2$ | $2$ | $0$ |
312.36.2.b.1 | $312$ | $3$ | $3$ | $2$ |
312.48.1.dh.1 | $312$ | $4$ | $4$ | $1$ |