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Properties

Label 40.96.1.l.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 $2^{4}\cdot8$
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: 40.96.1.783

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&0\\12&1\end{bmatrix}$, $\begin{bmatrix}5&18\\12&35\end{bmatrix}$, $\begin{bmatrix}21&36\\36&9\end{bmatrix}$, $\begin{bmatrix}29&30\\28&31\end{bmatrix}$, $\begin{bmatrix}39&14\\28&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.l.1.1, 40.192.1-40.l.1.2, 40.192.1-40.l.1.3, 40.192.1-40.l.1.4, 40.192.1-40.l.1.5, 40.192.1-40.l.1.6, 40.192.1-40.l.1.7, 40.192.1-40.l.1.8, 40.192.1-40.l.1.9, 40.192.1-40.l.1.10, 40.192.1-40.l.1.11, 40.192.1-40.l.1.12, 40.192.1-40.l.1.13, 40.192.1-40.l.1.14, 40.192.1-40.l.1.15, 40.192.1-40.l.1.16, 80.192.1-40.l.1.1, 80.192.1-40.l.1.2, 80.192.1-40.l.1.3, 80.192.1-40.l.1.4, 80.192.1-40.l.1.5, 80.192.1-40.l.1.6, 80.192.1-40.l.1.7, 80.192.1-40.l.1.8, 120.192.1-40.l.1.1, 120.192.1-40.l.1.2, 120.192.1-40.l.1.3, 120.192.1-40.l.1.4, 120.192.1-40.l.1.5, 120.192.1-40.l.1.6, 120.192.1-40.l.1.7, 120.192.1-40.l.1.8, 120.192.1-40.l.1.9, 120.192.1-40.l.1.10, 120.192.1-40.l.1.11, 120.192.1-40.l.1.12, 120.192.1-40.l.1.13, 120.192.1-40.l.1.14, 120.192.1-40.l.1.15, 120.192.1-40.l.1.16, 240.192.1-40.l.1.1, 240.192.1-40.l.1.2, 240.192.1-40.l.1.3, 240.192.1-40.l.1.4, 240.192.1-40.l.1.5, 240.192.1-40.l.1.6, 240.192.1-40.l.1.7, 240.192.1-40.l.1.8, 280.192.1-40.l.1.1, 280.192.1-40.l.1.2, 280.192.1-40.l.1.3, 280.192.1-40.l.1.4, 280.192.1-40.l.1.5, 280.192.1-40.l.1.6, 280.192.1-40.l.1.7, 280.192.1-40.l.1.8, 280.192.1-40.l.1.9, 280.192.1-40.l.1.10, 280.192.1-40.l.1.11, 280.192.1-40.l.1.12, 280.192.1-40.l.1.13, 280.192.1-40.l.1.14, 280.192.1-40.l.1.15, 280.192.1-40.l.1.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
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 y - z w $
$=$ $10 x^{2} - 5 y^{2} - 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 5 x^{4} + 50 x^{2} y^{2} + x^{2} z^{2} - 10 y^{2} z^{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 between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{5}z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

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

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.g.2 $8$ $2$ $2$ $1$ $0$ dimension zero
40.48.0.b.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.b.2 $40$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.192.5.a.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.192.5.b.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.192.5.g.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.192.5.j.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.192.5.o.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.192.5.r.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.192.5.w.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.192.5.x.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.480.33.cg.1 $40$ $5$ $5$ $33$ $5$ $1^{14}\cdot2^{9}$
40.576.33.ij.1 $40$ $6$ $6$ $33$ $2$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.le.1 $40$ $10$ $10$ $65$ $10$ $1^{28}\cdot2^{10}\cdot4^{4}$
80.192.5.cd.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.cd.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ce.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ce.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.9.hf.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.hf.2 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.hg.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.hg.2 $80$ $2$ $2$ $9$ $?$ not computed
120.192.5.bh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bi.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dm.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.er.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.es.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.gn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.go.1 $120$ $2$ $2$ $5$ $?$ not computed
120.288.17.brw.1 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.ow.1 $120$ $4$ $4$ $17$ $?$ not computed
240.192.5.gt.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.gt.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.gu.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.gu.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.baf.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.baf.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bag.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bag.2 $240$ $2$ $2$ $9$ $?$ not computed
280.192.5.ch.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.ci.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.dk.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.dl.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.eq.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.er.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fr.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fs.1 $280$ $2$ $2$ $5$ $?$ not computed