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Properties

Label 40.96.1.br.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 $2^{2}\cdot4^{3}$
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: 40.96.1.1074

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}13&32\\10&7\end{bmatrix}$, $\begin{bmatrix}17&8\\38&11\end{bmatrix}$, $\begin{bmatrix}23&0\\0&17\end{bmatrix}$, $\begin{bmatrix}31&8\\8&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.br.1.1, 40.192.1-40.br.1.2, 40.192.1-40.br.1.3, 40.192.1-40.br.1.4, 40.192.1-40.br.1.5, 40.192.1-40.br.1.6, 40.192.1-40.br.1.7, 40.192.1-40.br.1.8, 80.192.1-40.br.1.1, 80.192.1-40.br.1.2, 80.192.1-40.br.1.3, 80.192.1-40.br.1.4, 80.192.1-40.br.1.5, 80.192.1-40.br.1.6, 80.192.1-40.br.1.7, 80.192.1-40.br.1.8, 80.192.1-40.br.1.9, 80.192.1-40.br.1.10, 80.192.1-40.br.1.11, 80.192.1-40.br.1.12, 120.192.1-40.br.1.1, 120.192.1-40.br.1.2, 120.192.1-40.br.1.3, 120.192.1-40.br.1.4, 120.192.1-40.br.1.5, 120.192.1-40.br.1.6, 120.192.1-40.br.1.7, 120.192.1-40.br.1.8, 240.192.1-40.br.1.1, 240.192.1-40.br.1.2, 240.192.1-40.br.1.3, 240.192.1-40.br.1.4, 240.192.1-40.br.1.5, 240.192.1-40.br.1.6, 240.192.1-40.br.1.7, 240.192.1-40.br.1.8, 240.192.1-40.br.1.9, 240.192.1-40.br.1.10, 240.192.1-40.br.1.11, 240.192.1-40.br.1.12, 280.192.1-40.br.1.1, 280.192.1-40.br.1.2, 280.192.1-40.br.1.3, 280.192.1-40.br.1.4, 280.192.1-40.br.1.5, 280.192.1-40.br.1.6, 280.192.1-40.br.1.7, 280.192.1-40.br.1.8
Cyclic 40-isogeny field degree: $6$
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 $ $=$ $ 5 x^{2} + 5 y^{2} - z^{2} $
$=$ $10 x^{2} - 10 y^{2} - w^{2}$
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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 to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^2\,\frac{(16z^{8}+56z^{4}w^{4}+w^{8})^{3}}{w^{4}z^{4}(2z^{2}-w^{2})^{4}(2z^{2}+w^{2})^{4}}$

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.p.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.48.0.e.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.f.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.bc.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.bd.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.x.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.ds.1 $40$ $5$ $5$ $33$ $5$ $1^{14}\cdot2^{9}$
40.576.33.mm.1 $40$ $6$ $6$ $33$ $2$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.ry.2 $40$ $10$ $10$ $65$ $10$ $1^{28}\cdot2^{10}\cdot4^{4}$
80.192.5.e.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.z.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.cq.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.dk.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.do.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.dp.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.dq.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.dr.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.dy.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ed.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fd.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fy.2 $80$ $2$ $2$ $5$ $?$ not computed
120.288.17.coh.1 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.baq.1 $120$ $4$ $4$ $17$ $?$ not computed
240.192.5.m.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cv.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.jh.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.kg.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.kk.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.kl.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.km.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.kn.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.lk.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.lv.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.pj.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.rs.2 $240$ $2$ $2$ $5$ $?$ not computed