Invariants
| Level: | $32$ | $\SL_2$-level: | $32$ | Newform level: | $1024$ | ||
| Index: | $192$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $16^{4}\cdot32^{4}$ | Cusp orbits | $1^{2}\cdot2\cdot4$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $7$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 32G9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 32.192.9.119 |
Level structure
| $\GL_2(\Z/32\Z)$-generators: | $\begin{bmatrix}3&12\\30&1\end{bmatrix}$, $\begin{bmatrix}15&7\\10&17\end{bmatrix}$, $\begin{bmatrix}29&21\\26&3\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 32-isogeny field degree: | $8$ |
| Cyclic 32-torsion field degree: | $128$ |
| Full 32-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{81}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2\cdot4$ |
| Newforms: | 32.2.a.a, 256.2.a.b, 256.2.a.d, 1024.2.a.c, 1024.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ y v - z t $ |
| $=$ | $x r - y u$ | |
| $=$ | $x v + x s - y^{2} + r^{2}$ | |
| $=$ | $x z - x w + w u - v s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} y^{8} + 8 x^{6} y^{4} z^{4} - x^{6} z^{8} + 4 x^{4} y^{10} + 20 x^{4} y^{6} z^{4} + \cdots + 4 y^{2} z^{12} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1:0:0:0:0:0:0)$, $(0:0:1:1:0:0:0:0:0)$ |
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle r$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 16.96.3.ei.1 :
| $\displaystyle X$ | $=$ | $\displaystyle 2t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-r$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-r$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+X^{2}Y^{2}+2X^{2}YZ+Y^{3}Z+X^{2}Z^{2}+YZ^{3} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.3.ei.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $2\cdot4$ |
| 32.96.3.cc.1 | $32$ | $2$ | $2$ | $3$ | $3$ | $1^{2}\cdot4$ |
| 32.96.5.ca.1 | $32$ | $2$ | $2$ | $5$ | $5$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 32.384.25.bg.2 | $32$ | $2$ | $2$ | $25$ | $8$ | $1^{4}\cdot2^{4}\cdot4$ |
| 32.384.25.dt.1 | $32$ | $2$ | $2$ | $25$ | $13$ | $1^{4}\cdot2^{4}\cdot4$ |
| 32.384.25.ec.1 | $32$ | $2$ | $2$ | $25$ | $7$ | $1^{4}\cdot2^{4}\cdot4$ |
| 32.384.25.em.1 | $32$ | $2$ | $2$ | $25$ | $12$ | $1^{4}\cdot2^{4}\cdot4$ |