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: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 32C3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 32.96.3.62 |
Level structure
| $\GL_2(\Z/32\Z)$-generators: | $\begin{bmatrix}3&25\\24&31\end{bmatrix}$, $\begin{bmatrix}11&18\\8&5\end{bmatrix}$, $\begin{bmatrix}21&20\\0&5\end{bmatrix}$, $\begin{bmatrix}31&29\\24&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 32.48.3.d.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: | $1^{3}$ |
| Newforms: | 32.2.a.a, 128.2.a.a, 128.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x y w + y^{2} t - y z t $ |
| $=$ | $x z w + y z t - z^{2} t$ | |
| $=$ | $x w t + y t^{2} - z t^{2}$ | |
| $=$ | $x w^{2} + y w t - z w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{7} + 3 x^{4} y z^{2} + x y^{2} z^{4} - y z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -2x^{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.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(0:-1:1:0:0)$, $(0:0:1:0:0)$, $(0:1:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{xt^{6}-64y^{2}zt^{4}-12yz^{4}wt-24yz^{2}w^{2}t^{2}+70yz^{2}t^{4}+12ywt^{5}+2z^{7}-6z^{3}t^{4}-8zwt^{5}}{twz^{4}y}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 32.48.3.d.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{7}+3X^{4}YZ^{2}+XY^{2}Z^{4}-YZ^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 32.48.3.d.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2x^{4}-xzt^{2}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x$ |
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.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 32.48.1-16.b.1.6 | $32$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
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.c.2.9 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 32.192.5-32.h.1.5 | $32$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 32.192.5-32.i.2.9 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 32.192.5-32.m.1.5 | $32$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 32.192.5-32.v.1.7 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.v.2.6 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.y.1.8 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.y.2.5 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.be.1.3 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.be.2.7 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.bf.1.2 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.bf.2.3 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 96.192.5-96.bj.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bk.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bn.2.9 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bo.2.9 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cb.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cb.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.ce.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.ce.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.da.1.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.da.2.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.db.1.7 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.db.2.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.288.11-96.p.2.3 | $96$ | $3$ | $3$ | $11$ | $?$ | not computed |
| 96.384.13-96.ke.2.1 | $96$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 160.192.5-160.bj.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bk.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bn.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bo.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cb.1.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cb.2.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.ce.1.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.ce.2.11 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.da.1.6 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.da.2.14 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.db.1.4 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.db.2.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.480.19-160.h.2.3 | $160$ | $5$ | $5$ | $19$ | $?$ | not computed |
| 224.192.5-224.bj.2.3 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bk.2.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bn.2.9 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bo.2.9 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cb.1.10 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cb.2.9 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.ce.1.9 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.ce.2.11 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.da.1.3 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.da.2.7 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.db.1.4 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.db.2.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |