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Properties

Label 40.48.1.x.1
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: $64$
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: 40.48.1.296

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&16\\30&31\end{bmatrix}$, $\begin{bmatrix}27&36\\22&35\end{bmatrix}$, $\begin{bmatrix}29&36\\0&33\end{bmatrix}$, $\begin{bmatrix}33&20\\30&31\end{bmatrix}$, $\begin{bmatrix}39&12\\28&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.x.1.1, 40.96.1-40.x.1.2, 40.96.1-40.x.1.3, 40.96.1-40.x.1.4, 40.96.1-40.x.1.5, 40.96.1-40.x.1.6, 40.96.1-40.x.1.7, 40.96.1-40.x.1.8, 40.96.1-40.x.1.9, 40.96.1-40.x.1.10, 40.96.1-40.x.1.11, 40.96.1-40.x.1.12, 40.96.1-40.x.1.13, 40.96.1-40.x.1.14, 40.96.1-40.x.1.15, 40.96.1-40.x.1.16, 120.96.1-40.x.1.1, 120.96.1-40.x.1.2, 120.96.1-40.x.1.3, 120.96.1-40.x.1.4, 120.96.1-40.x.1.5, 120.96.1-40.x.1.6, 120.96.1-40.x.1.7, 120.96.1-40.x.1.8, 120.96.1-40.x.1.9, 120.96.1-40.x.1.10, 120.96.1-40.x.1.11, 120.96.1-40.x.1.12, 120.96.1-40.x.1.13, 120.96.1-40.x.1.14, 120.96.1-40.x.1.15, 120.96.1-40.x.1.16, 280.96.1-40.x.1.1, 280.96.1-40.x.1.2, 280.96.1-40.x.1.3, 280.96.1-40.x.1.4, 280.96.1-40.x.1.5, 280.96.1-40.x.1.6, 280.96.1-40.x.1.7, 280.96.1-40.x.1.8, 280.96.1-40.x.1.9, 280.96.1-40.x.1.10, 280.96.1-40.x.1.11, 280.96.1-40.x.1.12, 280.96.1-40.x.1.13, 280.96.1-40.x.1.14, 280.96.1-40.x.1.15, 280.96.1-40.x.1.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

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

$ 0 $ $=$ $ 2 x^{4} + 10 x^{2} y^{2} - 15 x^{2} z^{2} + 25 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{315y^{2}z^{10}-1890y^{2}z^{8}w^{2}+360y^{2}z^{6}w^{4}+720y^{2}z^{4}w^{6}-15120y^{2}z^{2}w^{8}+10080y^{2}w^{10}-31z^{12}+120z^{10}w^{2}+192z^{8}w^{4}-512z^{6}w^{6}+4080z^{4}w^{8}-6144z^{2}w^{10}+2048w^{12}}{w^{4}z^{4}(5y^{2}z^{2}+10y^{2}w^{2}-z^{4})}$

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
8.24.1.d.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.24.0.h.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.i.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.e.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.p.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.ba.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bg.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bi.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bo.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bq.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.br.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.bl.2 $40$ $5$ $5$ $17$ $2$ $1^{6}\cdot2^{5}$
40.288.17.ct.1 $40$ $6$ $6$ $17$ $0$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.gf.2 $40$ $10$ $10$ $33$ $4$ $1^{12}\cdot2^{6}\cdot4^{2}$
120.96.1.de.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dg.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.du.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ee.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ek.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.eu.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.fi.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.fk.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.144.9.ep.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.cv.2 $120$ $4$ $4$ $9$ $?$ not computed
280.96.1.dm.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.do.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.dw.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ec.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.em.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.es.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.fa.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.fc.2 $280$ $2$ $2$ $1$ $?$ dimension zero