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Properties

Label 136.24.0.bl.1
Level $136$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $136$ $\SL_2$-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

$\GL_2(\Z/136\Z)$-generators: $\begin{bmatrix}9&56\\90&77\end{bmatrix}$, $\begin{bmatrix}49&128\\0&23\end{bmatrix}$, $\begin{bmatrix}57&88\\93&81\end{bmatrix}$, $\begin{bmatrix}63&48\\71&47\end{bmatrix}$, $\begin{bmatrix}131&104\\113&97\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 136.48.0-136.bl.1.1, 136.48.0-136.bl.1.2, 136.48.0-136.bl.1.3, 136.48.0-136.bl.1.4, 136.48.0-136.bl.1.5, 136.48.0-136.bl.1.6, 136.48.0-136.bl.1.7, 136.48.0-136.bl.1.8, 136.48.0-136.bl.1.9, 136.48.0-136.bl.1.10, 136.48.0-136.bl.1.11, 136.48.0-136.bl.1.12, 272.48.0-136.bl.1.1, 272.48.0-136.bl.1.2, 272.48.0-136.bl.1.3, 272.48.0-136.bl.1.4, 272.48.0-136.bl.1.5, 272.48.0-136.bl.1.6, 272.48.0-136.bl.1.7, 272.48.0-136.bl.1.8
Cyclic 136-isogeny field degree: $18$
Cyclic 136-torsion field degree: $1152$
Full 136-torsion field degree: $5013504$

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
$X_0(8)$ $8$ $2$ $2$ $0$ $0$
136.12.0.s.1 $136$ $2$ $2$ $0$ $?$
136.12.0.bb.1 $136$ $2$ $2$ $0$ $?$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
136.48.0.bk.1 $136$ $2$ $2$ $0$
136.48.0.bk.2 $136$ $2$ $2$ $0$
136.48.0.bl.1 $136$ $2$ $2$ $0$
136.48.0.bl.2 $136$ $2$ $2$ $0$
272.48.0.bc.1 $272$ $2$ $2$ $0$
272.48.0.bc.2 $272$ $2$ $2$ $0$
272.48.0.bd.1 $272$ $2$ $2$ $0$
272.48.0.bd.2 $272$ $2$ $2$ $0$
272.48.1.r.1 $272$ $2$ $2$ $1$
272.48.1.t.1 $272$ $2$ $2$ $1$
272.48.1.ch.1 $272$ $2$ $2$ $1$
272.48.1.cj.1 $272$ $2$ $2$ $1$