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Properties

Label 40.96.1.x.1
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 $2^{6}\cdot4$
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: 40.96.1.176

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&8\\4&1\end{bmatrix}$, $\begin{bmatrix}17&4\\32&13\end{bmatrix}$, $\begin{bmatrix}23&24\\32&33\end{bmatrix}$, $\begin{bmatrix}33&8\\8&21\end{bmatrix}$, $\begin{bmatrix}39&8\\4&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.x.1.1, 40.192.1-40.x.1.2, 40.192.1-40.x.1.3, 40.192.1-40.x.1.4, 40.192.1-40.x.1.5, 40.192.1-40.x.1.6, 40.192.1-40.x.1.7, 40.192.1-40.x.1.8, 40.192.1-40.x.1.9, 40.192.1-40.x.1.10, 40.192.1-40.x.1.11, 40.192.1-40.x.1.12, 80.192.1-40.x.1.1, 80.192.1-40.x.1.2, 80.192.1-40.x.1.3, 80.192.1-40.x.1.4, 80.192.1-40.x.1.5, 80.192.1-40.x.1.6, 80.192.1-40.x.1.7, 80.192.1-40.x.1.8, 120.192.1-40.x.1.1, 120.192.1-40.x.1.2, 120.192.1-40.x.1.3, 120.192.1-40.x.1.4, 120.192.1-40.x.1.5, 120.192.1-40.x.1.6, 120.192.1-40.x.1.7, 120.192.1-40.x.1.8, 120.192.1-40.x.1.9, 120.192.1-40.x.1.10, 120.192.1-40.x.1.11, 120.192.1-40.x.1.12, 240.192.1-40.x.1.1, 240.192.1-40.x.1.2, 240.192.1-40.x.1.3, 240.192.1-40.x.1.4, 240.192.1-40.x.1.5, 240.192.1-40.x.1.6, 240.192.1-40.x.1.7, 240.192.1-40.x.1.8, 280.192.1-40.x.1.1, 280.192.1-40.x.1.2, 280.192.1-40.x.1.3, 280.192.1-40.x.1.4, 280.192.1-40.x.1.5, 280.192.1-40.x.1.6, 280.192.1-40.x.1.7, 280.192.1-40.x.1.8, 280.192.1-40.x.1.9, 280.192.1-40.x.1.10, 280.192.1-40.x.1.11, 280.192.1-40.x.1.12
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 $ $=$ $ x^{2} + 2 x y - z^{2} $
$=$ $x^{2} + 2 x y - 5 y^{2} + 4 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 5 x^{2} y^{2} - 6 x^{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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{5}w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{5^2}\cdot\frac{(10000z^{8}+4000z^{6}w^{2}+500z^{4}w^{4}+20z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(5z^{2}+w^{2})^{2}(10z^{2}+w^{2})^{4}}$

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.c.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.b.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.w.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.x.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.o.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.be.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bf.2 $40$ $2$ $2$ $1$ $1$ 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
40.192.5.x.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.192.5.y.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.192.5.ba.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.192.5.bb.3 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.480.33.cv.2 $40$ $5$ $5$ $33$ $4$ $1^{14}\cdot2^{9}$
40.576.33.jx.2 $40$ $6$ $6$ $33$ $4$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.nr.1 $40$ $10$ $10$ $65$ $6$ $1^{28}\cdot2^{10}\cdot4^{4}$
80.192.5.f.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.l.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bh.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bj.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.eo.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.eq.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fm.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fs.1 $80$ $2$ $2$ $5$ $?$ not computed
120.192.5.hi.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hk.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hs.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.288.17.cad.2 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.tn.2 $120$ $4$ $4$ $17$ $?$ not computed
240.192.5.bd.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bj.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.dt.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.dv.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.oe.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.og.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.qq.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.qw.1 $240$ $2$ $2$ $5$ $?$ not computed
280.192.5.hf.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.hg.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.hl.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.hm.1 $280$ $2$ $2$ $5$ $?$ not computed