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Properties

Label 40.96.1.z.2
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: $32$
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: 40.96.1.891

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&10\\32&39\end{bmatrix}$, $\begin{bmatrix}7&26\\24&25\end{bmatrix}$, $\begin{bmatrix}17&8\\0&21\end{bmatrix}$, $\begin{bmatrix}31&10\\4&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.z.2.1, 40.192.1-40.z.2.2, 40.192.1-40.z.2.3, 40.192.1-40.z.2.4, 40.192.1-40.z.2.5, 40.192.1-40.z.2.6, 40.192.1-40.z.2.7, 40.192.1-40.z.2.8, 80.192.1-40.z.2.1, 80.192.1-40.z.2.2, 80.192.1-40.z.2.3, 80.192.1-40.z.2.4, 80.192.1-40.z.2.5, 80.192.1-40.z.2.6, 80.192.1-40.z.2.7, 80.192.1-40.z.2.8, 80.192.1-40.z.2.9, 80.192.1-40.z.2.10, 80.192.1-40.z.2.11, 80.192.1-40.z.2.12, 120.192.1-40.z.2.1, 120.192.1-40.z.2.2, 120.192.1-40.z.2.3, 120.192.1-40.z.2.4, 120.192.1-40.z.2.5, 120.192.1-40.z.2.6, 120.192.1-40.z.2.7, 120.192.1-40.z.2.8, 240.192.1-40.z.2.1, 240.192.1-40.z.2.2, 240.192.1-40.z.2.3, 240.192.1-40.z.2.4, 240.192.1-40.z.2.5, 240.192.1-40.z.2.6, 240.192.1-40.z.2.7, 240.192.1-40.z.2.8, 240.192.1-40.z.2.9, 240.192.1-40.z.2.10, 240.192.1-40.z.2.11, 240.192.1-40.z.2.12, 280.192.1-40.z.2.1, 280.192.1-40.z.2.2, 280.192.1-40.z.2.3, 280.192.1-40.z.2.4, 280.192.1-40.z.2.5, 280.192.1-40.z.2.6, 280.192.1-40.z.2.7, 280.192.1-40.z.2.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $7680$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ 10 x^{2} + z^{2} - w^{2} $
$=$ $10 y^{2} + z^{2} - 2 w^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{(z^{8}-4z^{6}w^{2}+5z^{4}w^{4}-2z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(z-w)^{4}(z+w)^{4}(z^{2}-2w^{2})^{2}}$

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.k.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.48.0.o.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.p.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.s.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.t.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.q.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.r.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.cy.1 $40$ $5$ $5$ $33$ $5$ $1^{14}\cdot2^{9}$
40.576.33.kc.1 $40$ $6$ $6$ $33$ $5$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.ny.1 $40$ $10$ $10$ $65$ $11$ $1^{28}\cdot2^{10}\cdot4^{4}$
80.192.5.i.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.k.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bg.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bm.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.en.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.et.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fp.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fr.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.9.jt.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.ju.2 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.ka.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.kb.2 $80$ $2$ $2$ $9$ $?$ not computed
120.288.17.cal.1 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.tt.2 $120$ $4$ $4$ $17$ $?$ not computed
240.192.5.bg.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ds.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.dy.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.od.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.oj.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.qt.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.qv.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.bjr.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bjs.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bkg.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bkh.1 $240$ $2$ $2$ $9$ $?$ not computed