Invariants
| Level: | $171$ | $\SL_2$-level: | $9$ | Newform level: | $27$ | ||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $3\cdot9$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9A1 |
Level structure
| $\GL_2(\Z/171\Z)$-generators: | $\begin{bmatrix}12&32\\113&36\end{bmatrix}$, $\begin{bmatrix}29&88\\144&43\end{bmatrix}$, $\begin{bmatrix}119&51\\14&145\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.12.1.a.1 for the level structure with $-I$) |
| Cyclic 171-isogeny field degree: | $60$ |
| Cyclic 171-torsion field degree: | $6480$ |
| Full 171-torsion field degree: | $19945440$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 27.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{(y+z)^{2}(9y+z)^{3}}{z^{3}y(y+z)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 57.8.0-3.a.1.2 | $57$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 171.72.1-9.a.1.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-171.a.1.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-171.a.2.2 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-9.b.1.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-9.b.2.2 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-171.b.1.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-171.b.2.2 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-9.c.1.2 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-171.c.1.3 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.72.1-171.c.2.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.480.19-171.a.1.8 | $171$ | $20$ | $20$ | $19$ | $?$ | not computed |