Invariants
| Level: | $160$ | $\SL_2$-level: | $32$ | Newform level: | $128$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $3 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $4^{2}\cdot8\cdot32$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 32A3 |
Level structure
| $\GL_2(\Z/160\Z)$-generators: | $\begin{bmatrix}9&116\\14&135\end{bmatrix}$, $\begin{bmatrix}38&53\\95&148\end{bmatrix}$, $\begin{bmatrix}56&1\\37&20\end{bmatrix}$, $\begin{bmatrix}102&135\\119&54\end{bmatrix}$, $\begin{bmatrix}139&130\\18&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 32.48.3.a.1 for the level structure with $-I$) |
| Cyclic 160-isogeny field degree: | $24$ |
| Cyclic 160-torsion field degree: | $1536$ |
| Full 160-torsion field degree: | $1966080$ |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{3} z - x z^{3} + y^{4} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1)$, $(-1:0:1)$, $(1:0:0)$, $(1:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{(x^{4}-16x^{2}z^{2}+16z^{4})^{3}}{z^{2}x^{8}(x-z)(x+z)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 80.48.1-16.b.1.15 | $80$ | $2$ | $2$ | $1$ | $?$ |
| 160.48.1-16.b.1.1 | $160$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 160.192.5-32.d.1.11 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.g.2.5 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.j.1.10 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.l.1.5 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.r.2.6 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.r.1.10 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.s.1.5 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.s.1.10 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.t.1.4 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.t.1.14 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.u.2.3 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.u.1.14 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.v.1.3 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.w.2.8 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.x.2.2 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-32.y.1.6 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.z.2.4 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.ba.1.24 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.bb.2.6 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.bc.2.8 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.bd.2.4 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.be.2.4 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.bf.1.10 | $160$ | $2$ | $2$ | $5$ |
| 160.192.5-160.bg.2.2 | $160$ | $2$ | $2$ | $5$ |
| 160.480.19-160.a.1.22 | $160$ | $5$ | $5$ | $19$ |