Modular curve 40.48.1.fv.1

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

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:	8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	40.48.1.365

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}9&14\\27&39\end{bmatrix}$, $\begin{bmatrix}11&24\\24&7\end{bmatrix}$, $\begin{bmatrix}13&8\\31&3\end{bmatrix}$, $\begin{bmatrix}25&14\\12&7\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	40.96.1-40.fv.1.1, 40.96.1-40.fv.1.2, 120.96.1-40.fv.1.1, 120.96.1-40.fv.1.2, 280.96.1-40.fv.1.1, 280.96.1-40.fv.1.2
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 $	$=$	$ 2 y^{2} - 6 y z + y w + 2 z^{2} + z w $
	$=$	$5 x^{2} - y^{2} - 2 y z + 2 y w - z^{2} + 2 z w + w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{4} + 24 x^{3} z - 5 x^{2} y^{2} - 54 x^{2} z^{2} - 10 x y^{2} z + 24 x z^{3} - 5 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:

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

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle -\frac{2^2}{5^4}\cdot\frac{4608000000yz^{11}-25344000000yz^{10}w+60000000000yz^{9}w^{2}-79920000000yz^{8}w^{3}+65425440000yz^{7}w^{4}-33437040000yz^{6}w^{5}+10105480000yz^{5}w^{6}-1431100000yz^{4}w^{7}-32672200yz^{3}w^{8}+25588300yz^{2}w^{9}+709550yzw^{10}-202825yw^{11}-1760000000z^{12}+8256000000z^{11}w-15600000000z^{10}w^{2}+14560000000z^{9}w^{3}-5739120000z^{8}w^{4}-1116240000z^{7}w^{5}+2104480000z^{6}w^{6}-806300000z^{5}w^{7}+90097400z^{4}w^{8}+12401300z^{3}w^{9}-1318700z^{2}w^{10}-202825zw^{11}-131072w^{12}}{w^{4}(2688yz^{7}-9408yz^{6}w+12896yz^{5}w^{2}-8720yz^{4}w^{3}+3016yz^{3}w^{4}-508yz^{2}w^{5}+38yzw^{6}-yw^{7}-1024z^{8}+2752z^{7}w-2304z^{6}w^{2}+240z^{5}w^{3}+536z^{4}w^{4}-228z^{3}w^{5}+28z^{2}w^{6}-zw^{7})}$

Modular covers

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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.x.1	$8$	$2$	$2$	$1$	$0$	dimension zero
20.24.0.j.1	$20$	$2$	$2$	$0$	$0$	full Jacobian
40.24.0.ch.1	$40$	$2$	$2$	$0$	$0$	full Jacobian
40.24.0.dg.1	$40$	$2$	$2$	$0$	$0$	full Jacobian
40.24.0.dp.1	$40$	$2$	$2$	$0$	$0$	full Jacobian
40.24.1.be.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.24.1.bm.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.jz.1	$40$	$5$	$5$	$17$	$4$	$1^{14}\cdot2$
40.288.17.yf.1	$40$	$6$	$6$	$17$	$2$	$1^{14}\cdot2$
40.480.33.bpt.1	$40$	$10$	$10$	$33$	$7$	$1^{28}\cdot2^{2}$
120.144.9.ezz.1	$120$	$3$	$3$	$9$	$?$	not computed
120.192.9.bpv.1	$120$	$4$	$4$	$9$	$?$	not computed