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Properties

Label 168.12.0.bb.1
Level $168$
Index $12$
Genus $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $168$ $\SL_2$-level: $8$
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 $1^{2}\cdot2\cdot8$ 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: 8C0

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}25&74\\60&139\end{bmatrix}$, $\begin{bmatrix}32&125\\39&98\end{bmatrix}$, $\begin{bmatrix}43&128\\34&141\end{bmatrix}$, $\begin{bmatrix}70&33\\3&80\end{bmatrix}$, $\begin{bmatrix}99&98\\82&55\end{bmatrix}$, $\begin{bmatrix}109&80\\6&91\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.24.0-168.bb.1.1, 168.24.0-168.bb.1.2, 168.24.0-168.bb.1.3, 168.24.0-168.bb.1.4, 168.24.0-168.bb.1.5, 168.24.0-168.bb.1.6, 168.24.0-168.bb.1.7, 168.24.0-168.bb.1.8, 168.24.0-168.bb.1.9, 168.24.0-168.bb.1.10, 168.24.0-168.bb.1.11, 168.24.0-168.bb.1.12, 168.24.0-168.bb.1.13, 168.24.0-168.bb.1.14, 168.24.0-168.bb.1.15, 168.24.0-168.bb.1.16, 168.24.0-168.bb.1.17, 168.24.0-168.bb.1.18, 168.24.0-168.bb.1.19, 168.24.0-168.bb.1.20, 168.24.0-168.bb.1.21, 168.24.0-168.bb.1.22, 168.24.0-168.bb.1.23, 168.24.0-168.bb.1.24, 168.24.0-168.bb.1.25, 168.24.0-168.bb.1.26, 168.24.0-168.bb.1.27, 168.24.0-168.bb.1.28, 168.24.0-168.bb.1.29, 168.24.0-168.bb.1.30, 168.24.0-168.bb.1.31, 168.24.0-168.bb.1.32
Cyclic 168-isogeny field degree: $64$
Cyclic 168-torsion field degree: $3072$
Full 168-torsion field degree: $12386304$

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
$X_0(4)$ $4$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
168.24.0.x.1 $168$ $2$ $2$ $0$
168.24.0.ba.1 $168$ $2$ $2$ $0$
168.24.0.bn.1 $168$ $2$ $2$ $0$
168.24.0.bo.1 $168$ $2$ $2$ $0$
168.24.0.bq.1 $168$ $2$ $2$ $0$
168.24.0.bt.1 $168$ $2$ $2$ $0$
168.24.0.cd.1 $168$ $2$ $2$ $0$
168.24.0.ce.1 $168$ $2$ $2$ $0$
168.24.0.cg.1 $168$ $2$ $2$ $0$
168.24.0.cj.1 $168$ $2$ $2$ $0$
168.24.0.ct.1 $168$ $2$ $2$ $0$
168.24.0.cu.1 $168$ $2$ $2$ $0$
168.24.0.cw.1 $168$ $2$ $2$ $0$
168.24.0.cz.1 $168$ $2$ $2$ $0$
168.24.0.dz.1 $168$ $2$ $2$ $0$
168.24.0.ea.1 $168$ $2$ $2$ $0$
168.36.2.dh.1 $168$ $3$ $3$ $2$
168.48.1.zx.1 $168$ $4$ $4$ $1$
168.96.5.gb.1 $168$ $8$ $8$ $5$
168.252.16.dh.1 $168$ $21$ $21$ $16$
168.336.21.dh.1 $168$ $28$ $28$ $21$