Properties

Label 171.24.0-9.a.1.2
Level $171$
Index $24$
Genus $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $171$ $\SL_2$-level: $9$
Index: $24$ $\PSL_2$-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 $1^{3}\cdot9$ 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: 9B0

Level structure

$\GL_2(\Z/171\Z)$-generators: $\begin{bmatrix}57&5\\37&71\end{bmatrix}$, $\begin{bmatrix}113&157\\84&154\end{bmatrix}$, $\begin{bmatrix}164&63\\44&139\end{bmatrix}$
Contains $-I$: no $\quad$ (see 9.12.0.a.1 for the level structure with $-I$)
Cyclic 171-isogeny field degree: $20$
Cyclic 171-torsion field degree: $2160$
Full 171-torsion field degree: $19945440$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 3100 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 12 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{x^{15}(x^{3}-24y^{3})^{3}}{y^{9}x^{12}(x-3y)(x^{2}+3xy+9y^{2})}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
57.8.0-3.a.1.2 $57$ $3$ $3$ $0$ $0$

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

Covering curve Level Index Degree Genus
171.72.0-9.d.1.1 $171$ $3$ $3$ $0$
171.72.0-9.d.2.2 $171$ $3$ $3$ $0$
171.72.0-171.d.1.3 $171$ $3$ $3$ $0$
171.72.0-171.d.2.4 $171$ $3$ $3$ $0$
171.72.0-9.e.1.2 $171$ $3$ $3$ $0$
171.72.0-171.e.1.3 $171$ $3$ $3$ $0$
171.72.0-171.e.2.4 $171$ $3$ $3$ $0$
171.72.0-171.f.1.4 $171$ $3$ $3$ $0$
171.72.0-171.f.2.2 $171$ $3$ $3$ $0$
171.72.1-9.a.1.1 $171$ $3$ $3$ $1$
171.480.17-171.a.1.2 $171$ $20$ $20$ $17$