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Properties

Label 40.48.1.s.2
Level $40$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $32$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
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: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.284

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}11&4\\10&17\end{bmatrix}$, $\begin{bmatrix}15&4\\18&33\end{bmatrix}$, $\begin{bmatrix}19&32\\32&3\end{bmatrix}$, $\begin{bmatrix}21&12\\18&39\end{bmatrix}$, $\begin{bmatrix}21&28\\30&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.s.2.1, 40.96.1-40.s.2.2, 40.96.1-40.s.2.3, 40.96.1-40.s.2.4, 40.96.1-40.s.2.5, 40.96.1-40.s.2.6, 40.96.1-40.s.2.7, 40.96.1-40.s.2.8, 40.96.1-40.s.2.9, 40.96.1-40.s.2.10, 40.96.1-40.s.2.11, 40.96.1-40.s.2.12, 80.96.1-40.s.2.1, 80.96.1-40.s.2.2, 80.96.1-40.s.2.3, 80.96.1-40.s.2.4, 80.96.1-40.s.2.5, 80.96.1-40.s.2.6, 80.96.1-40.s.2.7, 80.96.1-40.s.2.8, 80.96.1-40.s.2.9, 80.96.1-40.s.2.10, 80.96.1-40.s.2.11, 80.96.1-40.s.2.12, 80.96.1-40.s.2.13, 80.96.1-40.s.2.14, 80.96.1-40.s.2.15, 80.96.1-40.s.2.16, 120.96.1-40.s.2.1, 120.96.1-40.s.2.2, 120.96.1-40.s.2.3, 120.96.1-40.s.2.4, 120.96.1-40.s.2.5, 120.96.1-40.s.2.6, 120.96.1-40.s.2.7, 120.96.1-40.s.2.8, 120.96.1-40.s.2.9, 120.96.1-40.s.2.10, 120.96.1-40.s.2.11, 120.96.1-40.s.2.12, 240.96.1-40.s.2.1, 240.96.1-40.s.2.2, 240.96.1-40.s.2.3, 240.96.1-40.s.2.4, 240.96.1-40.s.2.5, 240.96.1-40.s.2.6, 240.96.1-40.s.2.7, 240.96.1-40.s.2.8, 240.96.1-40.s.2.9, 240.96.1-40.s.2.10, 240.96.1-40.s.2.11, 240.96.1-40.s.2.12, 240.96.1-40.s.2.13, 240.96.1-40.s.2.14, 240.96.1-40.s.2.15, 240.96.1-40.s.2.16, 280.96.1-40.s.2.1, 280.96.1-40.s.2.2, 280.96.1-40.s.2.3, 280.96.1-40.s.2.4, 280.96.1-40.s.2.5, 280.96.1-40.s.2.6, 280.96.1-40.s.2.7, 280.96.1-40.s.2.8, 280.96.1-40.s.2.9, 280.96.1-40.s.2.10, 280.96.1-40.s.2.11, 280.96.1-40.s.2.12
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

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} + 5 x y - w^{2} $
$=$ $5 x y - 5 y^{2} + z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 50 x^{4} - 5 x^{2} y^{2} - 15 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^4\,\frac{315y^{2}z^{10}+945y^{2}z^{8}w^{2}+90y^{2}z^{6}w^{4}-90y^{2}z^{4}w^{6}-945y^{2}z^{2}w^{8}-315y^{2}w^{10}-31z^{12}-60z^{10}w^{2}+48z^{8}w^{4}+64z^{6}w^{6}+255z^{4}w^{8}+192z^{2}w^{10}+32w^{12}}{w^{4}z^{4}(5y^{2}z^{2}-5y^{2}w^{2}-z^{4})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.1.c.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.24.0.i.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.el.1 $40$ $2$ $2$ $0$ $0$ 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
40.96.1.q.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.t.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.u.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.v.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.be.2 $40$ $5$ $5$ $17$ $1$ $1^{6}\cdot2^{5}$
40.288.17.ch.1 $40$ $6$ $6$ $17$ $1$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.fo.2 $40$ $10$ $10$ $33$ $3$ $1^{12}\cdot2^{6}\cdot4^{2}$
80.96.3.bo.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.bp.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.bq.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.br.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.cq.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.cr.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.cs.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.ct.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.5.bz.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ca.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.cb.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.cc.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.cd.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ce.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.cf.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.cg.2 $80$ $2$ $2$ $5$ $?$ not computed
120.96.1.cm.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.cn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.co.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.cp.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.144.9.dv.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.cg.2 $120$ $4$ $4$ $9$ $?$ not computed
240.96.3.dk.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.dl.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.dm.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.dn.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.em.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.en.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.eo.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ep.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.gh.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gj.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gk.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gl.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gm.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gn.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.go.2 $240$ $2$ $2$ $5$ $?$ not computed
280.96.1.cu.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cv.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cw.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cx.2 $280$ $2$ $2$ $1$ $?$ dimension zero