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Properties

Label 40.96.1.bf.2
Level $40$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
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: $1$
$\Q$-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: 40.96.1.348

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&28\\20&31\end{bmatrix}$, $\begin{bmatrix}15&24\\16&23\end{bmatrix}$, $\begin{bmatrix}27&12\\18&31\end{bmatrix}$, $\begin{bmatrix}33&36\\26&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.bf.2.1, 40.192.1-40.bf.2.2, 40.192.1-40.bf.2.3, 40.192.1-40.bf.2.4, 40.192.1-40.bf.2.5, 40.192.1-40.bf.2.6, 40.192.1-40.bf.2.7, 40.192.1-40.bf.2.8, 120.192.1-40.bf.2.1, 120.192.1-40.bf.2.2, 120.192.1-40.bf.2.3, 120.192.1-40.bf.2.4, 120.192.1-40.bf.2.5, 120.192.1-40.bf.2.6, 120.192.1-40.bf.2.7, 120.192.1-40.bf.2.8, 280.192.1-40.bf.2.1, 280.192.1-40.bf.2.2, 280.192.1-40.bf.2.3, 280.192.1-40.bf.2.4, 280.192.1-40.bf.2.5, 280.192.1-40.bf.2.6, 280.192.1-40.bf.2.7, 280.192.1-40.bf.2.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $7680$

Jacobian

Conductor: $2^{5}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

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

$ 0 $ $=$ $ 4 x^{2} - 2 y^{2} + z^{2} $
$=$ $2 x^{2} + 4 y^{2} + 3 z^{2} - w^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{5^2}\cdot\frac{(625z^{8}-500z^{6}w^{2}+125z^{4}w^{4}-10z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(5z^{2}-2w^{2})^{2}(5z^{2}-w^{2})^{4}}$

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
8.48.0.e.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.f.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.v.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bd.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bf.1 $40$ $2$ $2$ $1$ $1$ 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.df.1 $40$ $5$ $5$ $33$ $8$ $1^{14}\cdot2^{9}$
40.576.33.kn.1 $40$ $6$ $6$ $33$ $7$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.on.2 $40$ $10$ $10$ $65$ $13$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.288.17.ccd.1 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.vz.1 $120$ $4$ $4$ $17$ $?$ not computed