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Properties

Label 104.24.0.bn.1
Level $104$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $104$ $\SL_2$-level: $8$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $2^{4}\cdot8^{2}$ Cusp orbits $1^{2}\cdot2^{2}$
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: 8G0

Level structure

$\GL_2(\Z/104\Z)$-generators: $\begin{bmatrix}13&16\\16&33\end{bmatrix}$, $\begin{bmatrix}25&40\\51&83\end{bmatrix}$, $\begin{bmatrix}39&88\\51&53\end{bmatrix}$, $\begin{bmatrix}43&80\\76&5\end{bmatrix}$, $\begin{bmatrix}101&72\\70&47\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 104.48.0-104.bn.1.1, 104.48.0-104.bn.1.2, 104.48.0-104.bn.1.3, 104.48.0-104.bn.1.4, 104.48.0-104.bn.1.5, 104.48.0-104.bn.1.6, 104.48.0-104.bn.1.7, 104.48.0-104.bn.1.8, 104.48.0-104.bn.1.9, 104.48.0-104.bn.1.10, 104.48.0-104.bn.1.11, 104.48.0-104.bn.1.12, 208.48.0-104.bn.1.1, 208.48.0-104.bn.1.2, 208.48.0-104.bn.1.3, 208.48.0-104.bn.1.4, 208.48.0-104.bn.1.5, 208.48.0-104.bn.1.6, 208.48.0-104.bn.1.7, 208.48.0-104.bn.1.8, 312.48.0-104.bn.1.1, 312.48.0-104.bn.1.2, 312.48.0-104.bn.1.3, 312.48.0-104.bn.1.4, 312.48.0-104.bn.1.5, 312.48.0-104.bn.1.6, 312.48.0-104.bn.1.7, 312.48.0-104.bn.1.8, 312.48.0-104.bn.1.9, 312.48.0-104.bn.1.10, 312.48.0-104.bn.1.11, 312.48.0-104.bn.1.12
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 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
$X_0(8)$ $8$ $2$ $2$ $0$ $0$
52.12.0.h.1 $52$ $2$ $2$ $0$ $0$
104.12.0.z.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
104.48.0.bm.1 $104$ $2$ $2$ $0$
104.48.0.bm.2 $104$ $2$ $2$ $0$
104.48.0.bn.1 $104$ $2$ $2$ $0$
104.48.0.bn.2 $104$ $2$ $2$ $0$
104.336.23.eg.1 $104$ $14$ $14$ $23$
208.48.0.be.1 $208$ $2$ $2$ $0$
208.48.0.be.2 $208$ $2$ $2$ $0$
208.48.0.bf.1 $208$ $2$ $2$ $0$
208.48.0.bf.2 $208$ $2$ $2$ $0$
208.48.1.u.1 $208$ $2$ $2$ $1$
208.48.1.w.1 $208$ $2$ $2$ $1$
208.48.1.ck.1 $208$ $2$ $2$ $1$
208.48.1.cm.1 $208$ $2$ $2$ $1$
312.48.0.dx.1 $312$ $2$ $2$ $0$
312.48.0.dx.2 $312$ $2$ $2$ $0$
312.48.0.dy.1 $312$ $2$ $2$ $0$
312.48.0.dy.2 $312$ $2$ $2$ $0$
312.72.4.jl.1 $312$ $3$ $3$ $4$
312.96.3.od.1 $312$ $4$ $4$ $3$