Modular curve 56.96.1.bm.1

• Label: 56.96.1.bm.1
• Level: $56$
• Index: $96$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$56$		$\SL_2$-level:	$8$		Newform level:	$64$
Index:	$96$		$\PSL_2$-index:	$96$
Genus:	$1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)		Cusp widths	$4^{8}\cdot8^{8}$		Cusp orbits	$2^{2}\cdot4^{3}$
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:	8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	56.96.1.997

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}9&44\\28&5\end{bmatrix}$, $\begin{bmatrix}17&28\\42&15\end{bmatrix}$, $\begin{bmatrix}41&48\\30&21\end{bmatrix}$, $\begin{bmatrix}47&52\\24&35\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.192.1-56.bm.1.1, 56.192.1-56.bm.1.2, 56.192.1-56.bm.1.3, 56.192.1-56.bm.1.4, 56.192.1-56.bm.1.5, 56.192.1-56.bm.1.6, 56.192.1-56.bm.1.7, 56.192.1-56.bm.1.8, 168.192.1-56.bm.1.1, 168.192.1-56.bm.1.2, 168.192.1-56.bm.1.3, 168.192.1-56.bm.1.4, 168.192.1-56.bm.1.5, 168.192.1-56.bm.1.6, 168.192.1-56.bm.1.7, 168.192.1-56.bm.1.8, 280.192.1-56.bm.1.1, 280.192.1-56.bm.1.2, 280.192.1-56.bm.1.3, 280.192.1-56.bm.1.4, 280.192.1-56.bm.1.5, 280.192.1-56.bm.1.6, 280.192.1-56.bm.1.7, 280.192.1-56.bm.1.8
Cyclic 56-isogeny field degree:	$16$
Cyclic 56-torsion field degree:	$384$
Full 56-torsion field degree:	$32256$

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 $	$=$	$ 7 x^{2} + z^{2} + w^{2} $
	$=$	$7 y^{2} + 2 z^{2} + 4 w^{2}$

Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2^8\,\frac{(z^{8}+4z^{6}w^{2}+5z^{4}w^{4}+2z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(z^{2}+w^{2})^{4}(z^{2}+2w^{2})^{2}}$

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
8.48.1.m.2	$8$	$2$	$2$	$1$	$0$	dimension zero
56.48.0.l.1	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.0.n.2	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.0.t.1	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.0.v.1	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.1.y.1	$56$	$2$	$2$	$1$	$0$	dimension zero
56.48.1.ba.1	$56$	$2$	$2$	$1$	$0$	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
56.768.49.hp.2	$56$	$8$	$8$	$49$	$8$	$1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.tg.2	$56$	$21$	$21$	$145$	$23$	$1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.ua.1	$56$	$28$	$28$	$193$	$31$	$1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
168.288.17.cjm.2	$168$	$3$	$3$	$17$	$?$	not computed
168.384.17.zr.2	$168$	$4$	$4$	$17$	$?$	not computed