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Properties

Label 40.96.1.f.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: 40.96.1.747

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&8\\28&27\end{bmatrix}$, $\begin{bmatrix}7&32\\8&5\end{bmatrix}$, $\begin{bmatrix}11&24\\20&17\end{bmatrix}$, $\begin{bmatrix}23&24\\20&37\end{bmatrix}$, $\begin{bmatrix}27&0\\20&37\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.f.2.1, 40.192.1-40.f.2.2, 40.192.1-40.f.2.3, 40.192.1-40.f.2.4, 40.192.1-40.f.2.5, 40.192.1-40.f.2.6, 40.192.1-40.f.2.7, 40.192.1-40.f.2.8, 40.192.1-40.f.2.9, 40.192.1-40.f.2.10, 40.192.1-40.f.2.11, 40.192.1-40.f.2.12, 80.192.1-40.f.2.1, 80.192.1-40.f.2.2, 80.192.1-40.f.2.3, 80.192.1-40.f.2.4, 80.192.1-40.f.2.5, 80.192.1-40.f.2.6, 80.192.1-40.f.2.7, 80.192.1-40.f.2.8, 120.192.1-40.f.2.1, 120.192.1-40.f.2.2, 120.192.1-40.f.2.3, 120.192.1-40.f.2.4, 120.192.1-40.f.2.5, 120.192.1-40.f.2.6, 120.192.1-40.f.2.7, 120.192.1-40.f.2.8, 120.192.1-40.f.2.9, 120.192.1-40.f.2.10, 120.192.1-40.f.2.11, 120.192.1-40.f.2.12, 240.192.1-40.f.2.1, 240.192.1-40.f.2.2, 240.192.1-40.f.2.3, 240.192.1-40.f.2.4, 240.192.1-40.f.2.5, 240.192.1-40.f.2.6, 240.192.1-40.f.2.7, 240.192.1-40.f.2.8, 280.192.1-40.f.2.1, 280.192.1-40.f.2.2, 280.192.1-40.f.2.3, 280.192.1-40.f.2.4, 280.192.1-40.f.2.5, 280.192.1-40.f.2.6, 280.192.1-40.f.2.7, 280.192.1-40.f.2.8, 280.192.1-40.f.2.9, 280.192.1-40.f.2.10, 280.192.1-40.f.2.11, 280.192.1-40.f.2.12
Cyclic 40-isogeny field degree: $6$
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 $ $=$ $ 5 x^{2} + z^{2} - w^{2} $
$=$ $5 y^{2} + z^{2} + w^{2}$
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Rational points

This modular curve has no real points, and therefore no 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 $X(1)$ :

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

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.h.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.48.0.a.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.b.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.bb.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.bc.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.m.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.q.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
40.192.5.j.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.k.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.192.5.k.3 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.480.33.ca.2 $40$ $5$ $5$ $33$ $3$ $1^{14}\cdot2^{9}$
40.576.33.hg.2 $40$ $6$ $6$ $33$ $1$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.jg.2 $40$ $10$ $10$ $65$ $7$ $1^{28}\cdot2^{10}\cdot4^{4}$
80.192.5.by.3 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.by.4 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bz.3 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bz.4 $80$ $2$ $2$ $5$ $?$ not computed
80.192.9.gl.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.gl.2 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.gm.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.gm.2 $80$ $2$ $2$ $9$ $?$ not computed
120.192.5.cj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.cj.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ck.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ck.4 $120$ $2$ $2$ $5$ $?$ not computed
120.288.17.blh.2 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.lv.2 $120$ $4$ $4$ $17$ $?$ not computed
240.192.5.gi.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.gi.4 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.gj.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.gj.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.uf.3 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.uf.4 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.ug.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.ug.2 $240$ $2$ $2$ $9$ $?$ not computed
280.192.5.bg.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.bg.4 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.bh.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.bh.4 $280$ $2$ $2$ $5$ $?$ not computed