Modular curve 40.48.1.co.1

• Label: 40.48.1.co.1
• Level: $40$
• Index: $48$
• Genus: $1$
• Analytic rank: $1$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$40$		$\SL_2$-level:	$8$		Newform level:	$800$
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	$2^{2}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-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:	40.48.1.183

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}1&30\\12&19\end{bmatrix}$, $\begin{bmatrix}25&8\\3&15\end{bmatrix}$, $\begin{bmatrix}27&36\\0&11\end{bmatrix}$, $\begin{bmatrix}31&12\\0&23\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	40.96.1-40.co.1.1, 40.96.1-40.co.1.2, 40.96.1-40.co.1.3, 40.96.1-40.co.1.4, 120.96.1-40.co.1.1, 120.96.1-40.co.1.2, 120.96.1-40.co.1.3, 120.96.1-40.co.1.4, 280.96.1-40.co.1.1, 280.96.1-40.co.1.2, 280.96.1-40.co.1.3, 280.96.1-40.co.1.4
Cyclic 40-isogeny field degree:	$24$
Cyclic 40-torsion field degree:	$384$
Full 40-torsion field degree:	$15360$

Jacobian

Conductor:	$2^{5}\cdot5^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	800.2.a.d

Models

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

$ 0 $	$=$	$ 2 x^{2} + 2 x y - 2 y^{2} - z^{2} $
	$=$	$3 x^{2} - 2 x y + 2 y^{2} + 3 z^{2} - 2 w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 81 x^{4} + 220 x^{2} y^{2} - 18 x^{2} z^{2} + 100 y^{4} - 30 y^{2} z^{2} + z^{4} $

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:

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle 2w$

Maps to other modular curves

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

$\displaystyle j$

$=$

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

Modular covers

Hi

Sorry, your browser does not support the nearby lattice.

Cover information Click on a modular curve in the diagram to see information about it.

See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.24.0.n.1	$8$	$2$	$2$	$0$	$0$	full Jacobian
40.24.0.s.1	$40$	$2$	$2$	$0$	$0$	full Jacobian
40.24.0.dt.1	$40$	$2$	$2$	$0$	$0$	full Jacobian
40.24.0.eb.1	$40$	$2$	$2$	$0$	$0$	full Jacobian
40.24.1.h.1	$40$	$2$	$2$	$1$	$1$	dimension zero
40.24.1.dj.1	$40$	$2$	$2$	$1$	$1$	dimension zero
40.24.1.dr.1	$40$	$2$	$2$	$1$	$1$	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.eh.1	$40$	$5$	$5$	$17$	$3$	$1^{14}\cdot2$
40.288.17.lq.1	$40$	$6$	$6$	$17$	$8$	$1^{14}\cdot2$
40.480.33.vh.1	$40$	$10$	$10$	$33$	$6$	$1^{28}\cdot2^{2}$
120.144.9.cfn.1	$120$	$3$	$3$	$9$	$?$	not computed
120.192.9.vn.1	$120$	$4$	$4$	$9$	$?$	not computed