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Properties

Label 32.384.5-32.o.1.4
Level $32$
Index $384$
Genus $5$
Analytic rank $0$
Cusps $24$
$\Q$-cusps $0$

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Invariants

Level: $32$ $\SL_2$-level: $32$ Newform level: $128$
Index: $384$ $\PSL_2$-index:$192$
Genus: $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps: $24$ (none of which are rational) Cusp widths $2^{8}\cdot4^{12}\cdot32^{4}$ Cusp orbits $2^{4}\cdot4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 32N5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 32.384.5.115

Level structure

$\GL_2(\Z/32\Z)$-generators: $\begin{bmatrix}11&23\\24&1\end{bmatrix}$, $\begin{bmatrix}23&17\\0&29\end{bmatrix}$, $\begin{bmatrix}31&17\\16&9\end{bmatrix}$
Contains $-I$: no $\quad$ (see 32.192.5.o.1 for the level structure with $-I$)
Cyclic 32-isogeny field degree: $4$
Cyclic 32-torsion field degree: $16$
Full 32-torsion field degree: $1024$

Jacobian

Conductor: $2^{32}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2^{2}$
Newforms: 64.2.a.a, 64.2.e.a, 128.2.e.b

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $ $=$ $ x^{2} - y z + w t $
$=$ $y z - y w + z t + w t$
$=$ $y^{2} + 2 y w - z^{2} + 2 z t - w^{2} + t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} y^{4} + 8 x^{4} y^{2} z^{2} + 4 x^{4} z^{4} - 8 x^{3} y^{5} + 8 x^{3} y^{4} z + 16 x^{3} y^{3} z^{2} + \cdots + z^{8} $
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Rational points

This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{12480yw^{23}-64110yw^{22}t-68746yw^{21}t^{2}+617648yw^{20}t^{3}+224664yw^{19}t^{4}-2439418yw^{18}t^{5}-799654yw^{17}t^{6}+4719008yw^{16}t^{7}+1101440yw^{15}t^{8}-6110012yw^{14}t^{9}-985940yw^{13}t^{10}+4926080yw^{12}t^{11}+287664yw^{11}t^{12}-3213860yw^{10}t^{13}+261380yw^{9}t^{14}+1426912yw^{8}t^{15}-399872yw^{7}t^{16}-621782yw^{6}t^{17}+130334yw^{5}t^{18}+146768yw^{4}t^{19}-26888yw^{3}t^{20}-14434yw^{2}t^{21}+3298ywt^{22}-4567z^{2}w^{22}+21516z^{2}w^{21}t+33339z^{2}w^{20}t^{2}-208256z^{2}w^{19}t^{3}-156801z^{2}w^{18}t^{4}+801660z^{2}w^{17}t^{5}+537365z^{2}w^{16}t^{6}-1459328z^{2}w^{15}t^{7}-616734z^{2}w^{14}t^{8}+2004536z^{2}w^{13}t^{9}+429158z^{2}w^{12}t^{10}-2136576z^{2}w^{11}t^{11}+429158z^{2}w^{10}t^{12}+2004536z^{2}w^{9}t^{13}-616734z^{2}w^{8}t^{14}-1459328z^{2}w^{7}t^{15}+537365z^{2}w^{6}t^{16}+801660z^{2}w^{5}t^{17}-156801z^{2}w^{4}t^{18}-208256z^{2}w^{3}t^{19}+33339z^{2}w^{2}t^{20}+21516z^{2}wt^{21}-4567z^{2}t^{22}+5836zw^{22}t-28598zw^{21}t^{2}-39790zw^{20}t^{3}+269744zw^{19}t^{4}+183268zw^{18}t^{5}-981538zw^{17}t^{6}-674858zw^{16}t^{7}+1491744zw^{15}t^{8}+972088zw^{14}t^{9}-795212zw^{13}t^{10}-1145980zw^{12}t^{11}-652928zw^{11}t^{12}+127624zw^{10}t^{13}+2100940zw^{9}t^{14}+132028zw^{8}t^{15}-1800352zw^{7}t^{16}-275076zw^{6}t^{17}+836098zw^{5}t^{18}+88938zw^{4}t^{19}-201136zw^{3}t^{20}+2068zw^{2}t^{21}+21078zwt^{22}-3346zt^{23}-4569w^{24}+21516w^{23}t+40487w^{22}t^{2}-241144w^{21}t^{3}-209445w^{20}t^{4}+1095428w^{19}t^{5}+788529w^{18}t^{6}-2470880w^{17}t^{7}-1545852w^{16}t^{8}+3370712w^{15}t^{9}+2113694w^{14}t^{10}-2935760w^{13}t^{11}-1645202w^{12}t^{12}+1205352w^{11}t^{13}+1067802w^{10}t^{14}-93152w^{9}t^{15}-391753w^{8}t^{16}-209892w^{7}t^{17}+94363w^{6}t^{18}+85512w^{5}t^{19}-19305w^{4}t^{20}-11372w^{3}t^{21}+2581w^{2}t^{22}-2t^{24}}{(w-t)^{2}(w+t)^{2}(4680yw^{19}+7934yw^{18}t+5042yw^{17}t^{2}+1420yw^{16}t^{3}-932yw^{15}t^{4}-2812yw^{14}t^{5}-5260yw^{13}t^{6}-5668yw^{12}t^{7}-6564yw^{11}t^{8}-7472yw^{10}t^{9}-3896yw^{9}t^{10}-2620yw^{8}t^{11}-1228yw^{7}t^{12}+1724yw^{6}t^{13}+2988yw^{5}t^{14}+4052yw^{4}t^{15}+4172yw^{3}t^{16}+3186yw^{2}t^{17}+1254ywt^{18}-1713z^{2}w^{18}-3628z^{2}w^{17}t-3621z^{2}w^{16}t^{2}-2520z^{2}w^{15}t^{3}-1200z^{2}w^{14}t^{4}+544z^{2}w^{13}t^{5}+1656z^{2}w^{12}t^{6}+3736z^{2}w^{11}t^{7}+4750z^{2}w^{10}t^{8}+3736z^{2}w^{9}t^{9}+4750z^{2}w^{8}t^{10}+3736z^{2}w^{7}t^{11}+1656z^{2}w^{6}t^{12}+544z^{2}w^{5}t^{13}-1200z^{2}w^{4}t^{14}-2520z^{2}w^{3}t^{15}-3621z^{2}w^{2}t^{16}-3628z^{2}wt^{17}-1713z^{2}t^{18}+2172zw^{18}t+4070zw^{17}t^{2}+3070zw^{16}t^{3}+988zw^{15}t^{4}-588zw^{14}t^{5}-2812zw^{13}t^{6}-2084zw^{12}t^{7}-4852zw^{11}t^{8}-5604zw^{10}t^{9}-2936zw^{8}t^{11}-1804zw^{7}t^{12}+1948zw^{6}t^{13}+1724zw^{5}t^{14}+3332zw^{4}t^{15}+3620zw^{3}t^{16}+2200zw^{2}t^{17}-678zwt^{18}-1254zt^{19}-1713w^{20}-3628w^{19}t-1113w^{18}t^{2}+2016w^{17}t^{3}+2376w^{16}t^{4}+2248w^{15}t^{5}+1432w^{14}t^{6}+2192w^{13}t^{7}+1014w^{12}t^{8}-960w^{11}t^{9}+1014w^{10}t^{10}-960w^{9}t^{11}-2080w^{8}t^{12}-1000w^{7}t^{13}-1424w^{6}t^{14}-816w^{5}t^{15}-45w^{4}t^{16}+908w^{3}t^{17}+795w^{2}t^{18})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 32.192.5.o.1 :

$\displaystyle X$ $=$ $\displaystyle x+y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ 4X^{4}Y^{4}+8X^{4}Y^{2}Z^{2}+4X^{4}Z^{4}-8X^{3}Y^{5}+8X^{3}Y^{4}Z+16X^{3}Y^{3}Z^{2}+16X^{3}Y^{2}Z^{3}+24X^{3}YZ^{4}+8X^{3}Z^{5}-4X^{2}Y^{6}-16X^{2}Y^{5}Z-12X^{2}Y^{4}Z^{2}+52X^{2}Y^{2}Z^{4}+16X^{2}YZ^{5}-4X^{2}Z^{6}-4XY^{7}-20XY^{6}Z-36XY^{5}Z^{2}-68XY^{4}Z^{3}-60XY^{3}Z^{4}-44XY^{2}Z^{5}-28XYZ^{6}+4XZ^{7}+Y^{8}-8Y^{6}Z^{2}-16Y^{5}Z^{3}-18Y^{4}Z^{4}-8Y^{2}Z^{6}+16YZ^{7}+Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.192.1-16.l.1.6 $16$ $2$ $2$ $1$ $0$ $2^{2}$
32.192.1-16.l.1.1 $32$ $2$ $2$ $1$ $0$ $2^{2}$
32.192.2-32.c.1.4 $32$ $2$ $2$ $2$ $0$ $1\cdot2$
32.192.2-32.c.1.6 $32$ $2$ $2$ $2$ $0$ $1\cdot2$
32.192.2-32.d.1.3 $32$ $2$ $2$ $2$ $0$ $1\cdot2$
32.192.2-32.d.1.6 $32$ $2$ $2$ $2$ $0$ $1\cdot2$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
32.768.13-32.bd.1.4 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.13-32.bf.1.2 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.13-32.bi.2.2 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.13-32.bj.1.2 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$