Invariants
| Level: | $171$ | $\SL_2$-level: | $9$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 3 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $3\cdot9$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $3$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9C0 |
Level structure
| $\GL_2(\Z/171\Z)$-generators: | $\begin{bmatrix}16&26\\97&48\end{bmatrix}$, $\begin{bmatrix}77&39\\140&94\end{bmatrix}$, $\begin{bmatrix}164&118\\161&111\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.12.0.b.1 for the level structure with $-I$) |
| Cyclic 171-isogeny field degree: | $60$ |
| Cyclic 171-torsion field degree: | $6480$ |
| 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 88 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x+3y)(x^{2}-3xy+9y^{2})(x^{3}+3y^{3})^{3}}{y^{9}x^{15}}$ |
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.f.1.2 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-9.f.2.1 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-9.g.1.2 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-171.g.1.1 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-171.g.2.2 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-171.h.1.1 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-171.h.2.2 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-171.i.1.3 | $171$ | $3$ | $3$ | $0$ |
| 171.72.0-171.i.2.1 | $171$ | $3$ | $3$ | $0$ |
| 171.72.1-9.a.1.1 | $171$ | $3$ | $3$ | $1$ |
| 171.480.17-171.b.1.8 | $171$ | $20$ | $20$ | $17$ |