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Properties

Label 40.96.1.bo.1
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 $2^{4}\cdot4^{2}$
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: 40.96.1.881

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}23&16\\20&13\end{bmatrix}$, $\begin{bmatrix}25&16\\12&21\end{bmatrix}$, $\begin{bmatrix}33&28\\16&15\end{bmatrix}$, $\begin{bmatrix}35&6\\8&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.bo.1.1, 40.192.1-40.bo.1.2, 40.192.1-40.bo.1.3, 40.192.1-40.bo.1.4, 40.192.1-40.bo.1.5, 40.192.1-40.bo.1.6, 40.192.1-40.bo.1.7, 40.192.1-40.bo.1.8, 120.192.1-40.bo.1.1, 120.192.1-40.bo.1.2, 120.192.1-40.bo.1.3, 120.192.1-40.bo.1.4, 120.192.1-40.bo.1.5, 120.192.1-40.bo.1.6, 120.192.1-40.bo.1.7, 120.192.1-40.bo.1.8, 280.192.1-40.bo.1.1, 280.192.1-40.bo.1.2, 280.192.1-40.bo.1.3, 280.192.1-40.bo.1.4, 280.192.1-40.bo.1.5, 280.192.1-40.bo.1.6, 280.192.1-40.bo.1.7, 280.192.1-40.bo.1.8
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 $ $=$ $ 3 y^{2} + y z + 2 y w + 2 z w + 2 w^{2} $
$=$ $10 x^{2} - 7 y^{2} + y z + 2 y w + z^{2} + 2 z w + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} + 56 x^{3} z - 10 x^{2} y^{2} + 24 x^{2} z^{2} - 20 x y^{2} z - 4 x z^{3} - 10 y^{2} z^{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:

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

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}{3^4\cdot5^4}\cdot\frac{282318251840yz^{23}+2831067301280yz^{22}w-22992674747200yz^{21}w^{2}-557172967404800yz^{20}w^{3}-4672481957900000yz^{19}w^{4}-19601887418240000yz^{18}w^{5}-29297641243080000yz^{17}w^{6}+121656274959120000yz^{16}w^{7}+941933991120000000yz^{15}w^{8}+3480127109360000000yz^{14}w^{9}+9085764649280000000yz^{13}w^{10}+17683593786560000000yz^{12}w^{11}+23965859616400000000yz^{11}w^{12}+15495201097600000000yz^{10}w^{13}-19375050692000000000yz^{9}w^{14}-75443699025600000000yz^{8}w^{15}-125563485120000000000yz^{7}w^{16}-140254505520000000000yz^{6}w^{17}-114136511200000000000yz^{5}w^{18}-68695609600000000000yz^{4}w^{19}-30041057200000000000yz^{3}w^{20}-9068998400000000000yz^{2}w^{21}-1693076000000000000yzw^{22}-147224000000000000yw^{23}+94143178827z^{24}+564636503680z^{23}w-14109045797520z^{22}w^{2}-215386480482400z^{21}w^{3}-1146039483131200z^{20}w^{4}-1393217912400000z^{19}w^{5}+20987793164980000z^{18}w^{6}+174511737627240000z^{17}w^{7}+745076838726810000z^{16}w^{8}+2034244623840000000z^{15}w^{9}+3480966754648000000z^{14}w^{10}+2207823316560000000z^{13}w^{11}-7834843141840000000z^{12}w^{12}-34962426659200000000z^{11}w^{13}-86098223655600000000z^{10}w^{14}-158954745164000000000z^{9}w^{15}-233375878465500000000z^{8}w^{16}-274773363840000000000z^{7}w^{17}-257430437160000000000z^{6}w^{18}-188942140400000000000z^{5}w^{19}-106098762240000000000z^{4}w^{20}-43955454400000000000z^{3}w^{21}-12650436400000000000z^{2}w^{22}-2257692000000000000zw^{23}-188141000000000000w^{24}}{z^{4}(yz^{19}-142yz^{18}w+2628yz^{17}w^{2}+161256yz^{16}w^{3}+2224770yz^{15}w^{4}+9178956yz^{14}w^{5}-98650272yz^{13}w^{6}-1738353192yz^{12}w^{7}-13116181950yz^{11}w^{8}-62940439100yz^{10}w^{9}-212340657100yz^{9}w^{10}-525569686200yz^{8}w^{11}-973681819500yz^{7}w^{12}-1359428028000yz^{6}w^{13}-1424981925000yz^{5}w^{14}-1104733590000yz^{4}w^{15}-614588250000yz^{3}w^{16}-232047000000yz^{2}w^{17}-53247500000yzw^{18}-5605000000yw^{19}+2z^{19}w-274z^{18}w^{2}+3936z^{17}w^{3}+335892z^{16}w^{4}+6159300z^{15}w^{5}+57747612z^{14}w^{6}+339114816z^{13}w^{7}+1419184029z^{12}w^{8}+4782179100z^{11}w^{9}+14285074520z^{10}w^{10}+38281770800z^{9}w^{11}+86760743550z^{8}w^{12}+156417756000z^{7}w^{13}+216165670500z^{6}w^{14}+223901025000z^{5}w^{15}+170118253125z^{4}w^{16}+91873500000z^{3}w^{17}+33380250000z^{2}w^{18}+7317500000zw^{19}+731750000w^{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.n.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.48.0.o.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.p.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.w.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.x.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.x.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.ba.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.do.1 $40$ $5$ $5$ $33$ $5$ $1^{14}\cdot2^{9}$
40.576.33.mg.2 $40$ $6$ $6$ $33$ $3$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.rq.1 $40$ $10$ $10$ $65$ $9$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.288.17.cni.1 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.zv.2 $120$ $4$ $4$ $17$ $?$ not computed