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Properties

Label 40.96.1.be.2
Level $40$
Index $96$
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: $96$ $\PSL_2$-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: 40.96.1.854

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}17&24\\22&5\end{bmatrix}$, $\begin{bmatrix}19&32\\4&33\end{bmatrix}$, $\begin{bmatrix}31&16\\32&23\end{bmatrix}$, $\begin{bmatrix}33&20\\24&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.be.2.1, 40.192.1-40.be.2.2, 40.192.1-40.be.2.3, 40.192.1-40.be.2.4, 40.192.1-40.be.2.5, 40.192.1-40.be.2.6, 40.192.1-40.be.2.7, 40.192.1-40.be.2.8, 120.192.1-40.be.2.1, 120.192.1-40.be.2.2, 120.192.1-40.be.2.3, 120.192.1-40.be.2.4, 120.192.1-40.be.2.5, 120.192.1-40.be.2.6, 120.192.1-40.be.2.7, 120.192.1-40.be.2.8, 280.192.1-40.be.2.1, 280.192.1-40.be.2.2, 280.192.1-40.be.2.3, 280.192.1-40.be.2.4, 280.192.1-40.be.2.5, 280.192.1-40.be.2.6, 280.192.1-40.be.2.7, 280.192.1-40.be.2.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $7680$

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 - 2 y w - z^{2} - 2 z w - 2 w^{2} $
$=$ $10 x^{2} - 5 y^{2} - z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 27 x^{4} - 4 x^{3} z - 75 x^{2} y^{2} - 2 x^{2} z^{2} + 4 x z^{3} + 50 y^{4} + 2 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

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

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}{5^2}\cdot\frac{5199247775077125yz^{23}+130598382998114350yz^{22}w+1639429754279890300yz^{21}w^{2}+13637973479828800200yz^{20}w^{3}+84147707197904250000yz^{19}w^{4}+408761977767803580000yz^{18}w^{5}+1620560871115505880000yz^{17}w^{6}+5367144671549897680000yz^{16}w^{7}+15082494144641080000000yz^{15}w^{8}+36340536118161360000000yz^{14}w^{9}+75582138556608672000000yz^{13}w^{10}+136211886798093504000000yz^{12}w^{11}+212994061907952000000000yz^{11}w^{12}+288722876164256000000000yz^{10}w^{13}+338200977315392000000000yz^{9}w^{14}+340402461021465600000000yz^{8}w^{15}+291882609651200000000000yz^{7}w^{16}+210605847372800000000000yz^{6}w^{17}+125656623667200000000000yz^{5}w^{18}+60450743193600000000000yz^{4}w^{19}+22575974400000000000000yz^{3}w^{20}+6154309632000000000000yz^{2}w^{21}+1092059136000000000000yzw^{22}+94961664000000000000yw^{23}+2599759999429883z^{24}+67033424982840950z^{23}w+865179272263947750z^{22}w^{2}+7411714640513913600z^{21}w^{3}+47175257792281246800z^{20}w^{4}+236853512494867260000z^{19}w^{5}+972660973418405940000z^{18}w^{6}+3345173034997444160000z^{17}w^{7}+9790233412430825040000z^{16}w^{8}+24650743878114640000000z^{15}w^{9}+53790259245003792000000z^{14}w^{10}+102183461764172544000000z^{13}w^{11}+169367607930407424000000z^{12}w^{12}+244981982567968000000000z^{11}w^{13}+308686917679584000000000z^{10}w^{14}+337534350252544000000000z^{9}w^{15}+318321514163008000000000z^{8}w^{16}+256613391820800000000000z^{7}w^{17}+174619597516800000000000z^{6}w^{18}+98544293990400000000000z^{5}w^{19}+44966121891840000000000z^{4}w^{20}+15971293184000000000000z^{3}w^{21}+4151654400000000000000z^{2}w^{22}+704274432000000000000zw^{23}+58689536000000000000w^{24}}{z^{4}(115964114805yz^{19}+3066606581142yz^{18}w+39032661820996yz^{17}w^{2}+317934468626264yz^{16}w^{3}+1858411904520000yz^{15}w^{4}+8284835821411200yz^{14}w^{5}+29220032419923200yz^{13}w^{6}+83460615987315200yz^{12}w^{7}+195982555650400000yz^{11}w^{8}+381835171156800000yz^{10}w^{9}+620081616767360000yz^{9}w^{10}+839785881227520000yz^{8}w^{11}+945180529280000000yz^{7}w^{12}+876984195840000000yz^{6}w^{13}+661602490880000000yz^{5}w^{14}+396977040384000000yz^{4}w^{15}+182912384000000000yz^{3}w^{16}+61005056000000000yz^{2}w^{17}+13161984000000000yzw^{18}+1385472000000000yw^{19}+57982060683z^{20}+1571958101454z^{19}w+20581484339230z^{18}w^{2}+173066193963552z^{17}w^{3}+1048421177367376z^{16}w^{4}+4864677807286400z^{15}w^{5}+17942739967401600z^{14}w^{6}+53882204978982400z^{13}w^{7}+133837962185065600z^{12}w^{8}+277776078443200000z^{11}w^{9}+484536908512960000z^{10}w^{10}+711919870430720000z^{9}w^{11}+879986773285120000z^{8}w^{12}+910675339520000000z^{7}w^{13}+781900117760000000z^{6}w^{14}+548891996160000000z^{5}w^{15}+307984120320000000z^{4}w^{16}+133291776000000000z^{3}w^{17}+41925376000000000z^{2}w^{18}+8562688000000000zw^{19}+856268800000000w^{20})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.1.m.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.48.0.j.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.l.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.r.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.t.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.u.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.y.1 $40$ $2$ $2$ $1$ $0$ 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
40.480.33.de.1 $40$ $5$ $5$ $33$ $7$ $1^{14}\cdot2^{9}$
40.576.33.km.1 $40$ $6$ $6$ $33$ $3$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.om.2 $40$ $10$ $10$ $65$ $11$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.288.17.ccc.1 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.vy.1 $120$ $4$ $4$ $17$ $?$ not computed