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 | $171$ | $20$ | $20$ | $17$ |