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Properties

Label 40.192.1-40.e.2.4
Level $40$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $64$
Index: $192$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{2}\cdot4^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.192.1.892

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&8\\12&33\end{bmatrix}$, $\begin{bmatrix}21&12\\20&23\end{bmatrix}$, $\begin{bmatrix}31&20\\0&29\end{bmatrix}$, $\begin{bmatrix}33&36\\4&37\end{bmatrix}$
Contains $-I$: no $\quad$ (see 40.96.1.e.2 for the level structure with $-I$)
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $3840$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 y^{2} + y z + y w - 2 z w $
$=$ $10 x^{2} + y^{2} - 2 y z - 2 y w + 2 z^{2} + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 26 x^{4} - 4 x^{3} z + 5 x^{2} y^{2} + 6 x^{2} z^{2} - 10 x y^{2} z - 4 x z^{3} + 5 y^{2} z^{2} + z^{4} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{5^4}\cdot\frac{15483269538945yz^{23}+268265259438255yz^{22}w+2079835074643605yz^{21}w^{2}+11903021725807995yz^{20}w^{3}+55833569988462375yz^{19}w^{4}+211390644134005545yz^{18}w^{5}+676291511427135755yz^{17}w^{6}+1859413674911768805yz^{16}w^{7}+4301166580908770570yz^{15}w^{8}+8403809385708865750yz^{14}w^{9}+13809291204858267570yz^{13}w^{10}+18149368822733294830yz^{12}w^{11}+18149368822733294830yz^{11}w^{12}+13809291204858267570yz^{10}w^{13}+8403809385708865750yz^{9}w^{14}+4301166580908770570yz^{8}w^{15}+1859413674911768805yz^{7}w^{16}+676291511427135755yz^{6}w^{17}+211390644134005545yz^{5}w^{18}+55833569988462375yz^{4}w^{19}+11903021725807995yz^{3}w^{20}+2079835074643605yz^{2}w^{21}+268265259438255yzw^{22}+15483269538945yw^{23}-34359738368z^{24}-30141905357058z^{23}w-395304279880788z^{22}w^{2}-2669026307619758z^{21}w^{3}-14428750181859768z^{20}w^{4}-62912135608399878z^{19}w^{5}-222095419696400068z^{18}w^{6}-673283807169269098z^{17}w^{7}-1734107256534187168z^{16}w^{8}-3730157454314875828z^{15}w^{9}-6795175332680882408z^{14}w^{10}-10203098053887035948z^{13}w^{11}-11812830566148987728z^{12}w^{12}-10203098053887035948z^{11}w^{13}-6795175332680882408z^{10}w^{14}-3730157454314875828z^{9}w^{15}-1734107256534187168z^{8}w^{16}-673283807169269098z^{7}w^{17}-222095419696400068z^{6}w^{18}-62912135608399878z^{5}w^{19}-14428750181859768z^{4}w^{20}-2669026307619758z^{3}w^{21}-395304279880788z^{2}w^{22}-30141905357058zw^{23}-34359738368w^{24}}{(z-w)^{4}(2313441yz^{19}+117510939yz^{18}w+2402989875yz^{17}w^{2}+26208643305yz^{16}w^{3}+172781347300yz^{15}w^{4}+748902100428yz^{14}w^{5}+2273224195932yz^{13}w^{6}+5054979150100yz^{12}w^{7}+8484784435774yz^{11}w^{8}+10946037312906yz^{10}w^{9}+10946037312906yz^{9}w^{10}+8484784435774yz^{8}w^{11}+5054979150100yz^{7}w^{12}+2273224195932yz^{6}w^{13}+748902100428yz^{5}w^{14}+172781347300yz^{4}w^{15}+26208643305yz^{3}w^{16}+2402989875yz^{2}w^{17}+117510939yzw^{18}+2313441yw^{19}-4626882z^{19}w-211887468z^{18}w^{2}-3839080050z^{17}w^{3}-36626796960z^{16}w^{4}-210416868200z^{15}w^{5}-797803810256z^{14}w^{6}-2130816443784z^{13}w^{7}-4186740446048z^{12}w^{8}-6217600364252z^{11}w^{9}-7082631352200z^{10}w^{10}-6217600364252z^{9}w^{11}-4186740446048z^{8}w^{12}-2130816443784z^{7}w^{13}-797803810256z^{6}w^{14}-210416868200z^{5}w^{15}-36626796960z^{4}w^{16}-3839080050z^{3}w^{17}-211887468z^{2}w^{18}-4626882zw^{19})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.e.2 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle 2w$

Equation of the image curve:

$0$ $=$ $ 26X^{4}+5X^{2}Y^{2}-4X^{3}Z-10XY^{2}Z+6X^{2}Z^{2}+5Y^{2}Z^{2}-4XZ^{3}+Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.1-8.g.1.12 $8$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-8.g.1.10 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.0-40.a.1.5 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.a.1.14 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.b.2.10 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.b.2.18 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.y.1.4 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.y.1.11 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.z.2.8 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.z.2.15 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.1-40.w.1.8 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.w.1.11 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.x.2.4 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.x.2.9 $40$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.384.5-40.g.1.6 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.384.5-40.h.1.6 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.384.5-40.i.2.8 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.384.5-40.j.3.6 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.960.33-40.bz.2.8 $40$ $5$ $5$ $33$ $5$ $1^{14}\cdot2^{9}$
40.1152.33-40.hf.1.9 $40$ $6$ $6$ $33$ $2$ $1^{14}\cdot2\cdot4^{4}$
40.1920.65-40.jf.2.11 $40$ $10$ $10$ $65$ $10$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.384.5-120.cf.2.16 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.cg.2.16 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ch.2.16 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ci.2.16 $120$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.bc.2.16 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.bd.2.16 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.be.2.16 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.bf.2.16 $280$ $2$ $2$ $5$ $?$ not computed