Properties

Label 168.48.0-84.p.1.7
Level $168$
Index $48$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $168$ $\SL_2$-level: $6$
Index: $48$ $\PSL_2$-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 $2^{3}\cdot6^{3}$ 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: 6I0

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}89&34\\120&97\end{bmatrix}$, $\begin{bmatrix}95&6\\108&77\end{bmatrix}$, $\begin{bmatrix}122&13\\3&4\end{bmatrix}$, $\begin{bmatrix}154&81\\25&44\end{bmatrix}$, $\begin{bmatrix}163&132\\52&101\end{bmatrix}$
Contains $-I$: no $\quad$ (see 84.24.0.p.1 for the level structure with $-I$)
Cyclic 168-isogeny field degree: $32$
Cyclic 168-torsion field degree: $1536$
Full 168-torsion field degree: $3096576$

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
24.24.0-6.a.1.11 $24$ $2$ $2$ $0$ $0$
168.16.0-84.b.1.4 $168$ $3$ $3$ $0$ $?$
168.24.0-6.a.1.2 $168$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
168.96.1-84.n.1.1 $168$ $2$ $2$ $1$
168.96.1-84.p.1.2 $168$ $2$ $2$ $1$
168.96.1-84.bc.1.5 $168$ $2$ $2$ $1$
168.96.1-84.bf.1.5 $168$ $2$ $2$ $1$
168.96.1-84.bk.1.1 $168$ $2$ $2$ $1$
168.96.1-84.bn.1.1 $168$ $2$ $2$ $1$
168.96.1-84.bt.1.5 $168$ $2$ $2$ $1$
168.96.1-84.bv.1.9 $168$ $2$ $2$ $1$
168.96.1-168.zk.1.13 $168$ $2$ $2$ $1$
168.96.1-168.zq.1.12 $168$ $2$ $2$ $1$
168.96.1-168.blh.1.13 $168$ $2$ $2$ $1$
168.96.1-168.blq.1.14 $168$ $2$ $2$ $1$
168.96.1-168.byv.1.11 $168$ $2$ $2$ $1$
168.96.1-168.bze.1.14 $168$ $2$ $2$ $1$
168.96.1-168.bzv.1.13 $168$ $2$ $2$ $1$
168.96.1-168.cab.1.14 $168$ $2$ $2$ $1$
168.144.1-84.u.1.6 $168$ $3$ $3$ $1$
168.384.11-84.cl.1.13 $168$ $8$ $8$ $11$