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Properties

Label 40.48.1.ea.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: $1600$
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: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.3

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}11&8\\32&39\end{bmatrix}$, $\begin{bmatrix}13&3\\12&35\end{bmatrix}$, $\begin{bmatrix}39&4\\8&7\end{bmatrix}$, $\begin{bmatrix}39&37\\24&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.ea.1.1, 40.96.1-40.ea.1.2, 40.96.1-40.ea.1.3, 40.96.1-40.ea.1.4, 80.96.1-40.ea.1.1, 80.96.1-40.ea.1.2, 80.96.1-40.ea.1.3, 80.96.1-40.ea.1.4, 120.96.1-40.ea.1.1, 120.96.1-40.ea.1.2, 120.96.1-40.ea.1.3, 120.96.1-40.ea.1.4, 240.96.1-40.ea.1.1, 240.96.1-40.ea.1.2, 240.96.1-40.ea.1.3, 240.96.1-40.ea.1.4, 280.96.1-40.ea.1.1, 280.96.1-40.ea.1.2, 280.96.1-40.ea.1.3, 280.96.1-40.ea.1.4
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{6}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1600.2.a.n

Models

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

$ 0 $ $=$ $ 2 x^{2} + 2 x y - 4 x z + y^{2} $
$=$ $5 y z - 5 z^{2} + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 25 x^{4} - 10 x^{3} y + 2 x^{2} y^{2} - 20 x^{2} z^{2} - 4 x y z^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 5x$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{5^2}\cdot\frac{(5y^{2}-4w^{2})^{3}(5y^{2}+4w^{2})^{3}}{w^{8}y^{4}}$

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
4.24.0.c.1 $4$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.bf.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.ea.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.eb.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.n.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.dm.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.dn.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
40.240.17.hw.1 $40$ $5$ $5$ $17$ $4$ $1^{14}\cdot2$
40.288.17.vo.1 $40$ $6$ $6$ $17$ $6$ $1^{14}\cdot2$
40.480.33.bka.1 $40$ $10$ $10$ $33$ $8$ $1^{28}\cdot2^{2}$
80.96.3.ht.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.hx.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.ld.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.lh.1 $80$ $2$ $2$ $3$ $?$ not computed
120.144.9.dzc.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.bjm.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.3.wb.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.wf.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bfp.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bft.1 $240$ $2$ $2$ $3$ $?$ not computed