Invariants
| Level: | $32$ | $\SL_2$-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: | $0$ | ||||||
| $\Q$-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 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 32.96.3.42 |
Level structure
| $\GL_2(\Z/32\Z)$-generators: | $\begin{bmatrix}5&2\\16&9\end{bmatrix}$, $\begin{bmatrix}13&20\\16&5\end{bmatrix}$, $\begin{bmatrix}21&6\\8&19\end{bmatrix}$, $\begin{bmatrix}23&29\\8&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 32.48.3.a.2 for the level structure with $-I$) |
| Cyclic 32-isogeny field degree: | $4$ |
| Cyclic 32-torsion field degree: | $32$ |
| Full 32-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{19}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 32.2.a.a, 128.2.b.b |
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 |
|---|
| $(1:0:1)$, $(0:0:1)$, $(-1:0:1)$, $(1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{(x^{4}+60x^{3}z+134x^{2}z^{2}+60xz^{3}+z^{4})^{3}}{zx(x-z)^{8}(x+z)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.1-16.b.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 32.48.1-16.b.1.1 | $32$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 32.192.5-32.d.2.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 32.192.5-32.g.1.7 | $32$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 32.192.5-32.j.2.11 | $32$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 32.192.5-32.l.2.6 | $32$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 32.192.5-32.r.1.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 32.192.5-32.s.2.10 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 32.192.5-32.t.2.7 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 32.192.5-32.u.1.7 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 32.192.5-32.v.2.5 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 32.192.5-32.w.1.15 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 32.192.5-32.x.1.7 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 32.192.5-32.y.2.6 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 96.192.5-96.r.1.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.s.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.t.2.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.u.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.z.1.16 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.ba.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bb.1.16 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bc.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bd.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.be.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bf.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bg.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.288.11-96.a.1.62 | $96$ | $3$ | $3$ | $11$ | $?$ | not computed |
| 96.384.13-96.jt.1.61 | $96$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 160.192.5-160.r.2.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.s.2.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.t.2.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.u.2.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.z.1.16 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.ba.2.20 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bb.1.16 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bc.1.14 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bd.1.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.be.1.26 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bf.2.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bg.1.12 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.480.19-160.a.2.20 | $160$ | $5$ | $5$ | $19$ | $?$ | not computed |
| 224.192.5-224.r.1.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.s.1.14 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.t.2.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.u.2.12 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.z.1.16 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.ba.2.12 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bb.1.16 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bc.1.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bd.1.11 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.be.1.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bf.2.11 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bg.1.12 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |