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: | 32.96.3.60 |
Level structure
| $\GL_2(\Z/32\Z)$-generators: | $\begin{bmatrix}3&22\\24&29\end{bmatrix}$, $\begin{bmatrix}11&8\\24&1\end{bmatrix}$, $\begin{bmatrix}15&4\\8&9\end{bmatrix}$, $\begin{bmatrix}31&29\\8&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 32.48.3.d.1 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^{2} t + w t^{2} $ |
| $=$ | $x^{2} w + w^{2} t$ | |
| $=$ | $x^{2} z + z w t$ | |
| $=$ | $x^{2} y + y w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{7} + 3 x^{4} y z^{2} + x y^{2} z^{4} + 4 y z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{4} y $ | $=$ | $ 6x^{4} + 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{4xt^{6}-128y^{2}zt^{4}-12yz^{4}wt+48yz^{2}w^{2}t^{2}+140yz^{2}t^{4}-48ywt^{5}-z^{7}-12z^{3}t^{4}+32zwt^{5}}{twz^{4}y}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 32.48.3.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 8z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{7}+3X^{4}YZ^{2}+XY^{2}Z^{4}+4YZ^{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.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -6x^{4}-8xzt^{2}-t^{4}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x$ |
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.1.7 | $32$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
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.1.9 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 32.192.5-32.h.1.5 | $32$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 32.192.5-32.i.1.9 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 32.192.5-32.m.1.5 | $32$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 32.192.5-32.w.1.11 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.w.2.12 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.x.1.8 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.x.2.3 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.bd.1.1 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.bd.2.6 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.bg.1.3 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 32.192.5-32.bg.2.3 | $32$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 64.192.7-64.b.1.4 | $64$ | $2$ | $2$ | $7$ | $1$ | $4$ |
| 64.192.7-64.b.2.4 | $64$ | $2$ | $2$ | $7$ | $1$ | $4$ |
| 64.192.7-64.d.1.11 | $64$ | $2$ | $2$ | $7$ | $3$ | $2^{2}$ |
| 64.192.7-64.d.2.13 | $64$ | $2$ | $2$ | $7$ | $3$ | $2^{2}$ |
| 64.192.7-64.f.1.6 | $64$ | $2$ | $2$ | $7$ | $3$ | $2^{2}$ |
| 64.192.7-64.f.2.3 | $64$ | $2$ | $2$ | $7$ | $3$ | $2^{2}$ |
| 64.192.7-64.h.1.6 | $64$ | $2$ | $2$ | $7$ | $1$ | $4$ |
| 64.192.7-64.h.2.3 | $64$ | $2$ | $2$ | $7$ | $1$ | $4$ |
| 96.192.5-96.bj.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bk.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bn.1.9 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bo.1.9 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cc.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cc.2.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cd.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cd.2.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cz.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cz.2.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.dc.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.dc.2.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.288.11-96.p.1.17 | $96$ | $3$ | $3$ | $11$ | $?$ | not computed |
| 96.384.13-96.ke.1.33 | $96$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 160.192.5-160.bj.1.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bk.1.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bn.1.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bo.1.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cc.1.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cc.2.21 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cd.1.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cd.2.11 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cz.1.11 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cz.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.dc.1.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.dc.2.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.480.19-160.h.1.19 | $160$ | $5$ | $5$ | $19$ | $?$ | not computed |
| 192.192.7-192.b.1.11 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 192.192.7-192.b.2.7 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 192.192.7-192.d.1.2 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 192.192.7-192.d.2.6 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 192.192.7-192.f.1.15 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 192.192.7-192.f.2.11 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 192.192.7-192.h.1.11 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 192.192.7-192.h.2.3 | $192$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 224.192.5-224.bj.1.3 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bk.1.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bn.1.9 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bo.1.9 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cc.1.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cc.2.13 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cd.1.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cd.2.13 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cz.1.11 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cz.2.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.dc.1.10 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.dc.2.3 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 320.192.7-320.b.1.5 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 320.192.7-320.b.2.7 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 320.192.7-320.d.1.10 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 320.192.7-320.d.2.18 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 320.192.7-320.f.1.11 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 320.192.7-320.f.2.5 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 320.192.7-320.h.1.11 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 320.192.7-320.h.2.3 | $320$ | $2$ | $2$ | $7$ | $?$ | not computed |