Modular curve 56.48.1.bi.2

• Label: 56.48.1.bi.2
• Level: $56$
• Index: $48$
• Genus: $1$
• Analytic rank: $1$
• Cusps: $8$
• $\Q$-cusps: $2$

Invariants

Level:	$56$		$\SL_2$-level:	$8$		Newform level:	$3136$
Index:	$48$		$\PSL_2$-index:	$48$
Genus:	$1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $2$ are rational)		Cusp widths	$4^{4}\cdot8^{4}$		Cusp orbits	$1^{2}\cdot2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	56.48.1.64

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}17&8\\46&37\end{bmatrix}$, $\begin{bmatrix}17&12\\12&37\end{bmatrix}$, $\begin{bmatrix}33&28\\8&1\end{bmatrix}$, $\begin{bmatrix}47&8\\28&53\end{bmatrix}$, $\begin{bmatrix}55&40\\22&19\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.96.1-56.bi.2.1, 56.96.1-56.bi.2.2, 56.96.1-56.bi.2.3, 56.96.1-56.bi.2.4, 56.96.1-56.bi.2.5, 56.96.1-56.bi.2.6, 56.96.1-56.bi.2.7, 56.96.1-56.bi.2.8, 56.96.1-56.bi.2.9, 56.96.1-56.bi.2.10, 56.96.1-56.bi.2.11, 56.96.1-56.bi.2.12, 56.96.1-56.bi.2.13, 56.96.1-56.bi.2.14, 56.96.1-56.bi.2.15, 56.96.1-56.bi.2.16, 168.96.1-56.bi.2.1, 168.96.1-56.bi.2.2, 168.96.1-56.bi.2.3, 168.96.1-56.bi.2.4, 168.96.1-56.bi.2.5, 168.96.1-56.bi.2.6, 168.96.1-56.bi.2.7, 168.96.1-56.bi.2.8, 168.96.1-56.bi.2.9, 168.96.1-56.bi.2.10, 168.96.1-56.bi.2.11, 168.96.1-56.bi.2.12, 168.96.1-56.bi.2.13, 168.96.1-56.bi.2.14, 168.96.1-56.bi.2.15, 168.96.1-56.bi.2.16, 280.96.1-56.bi.2.1, 280.96.1-56.bi.2.2, 280.96.1-56.bi.2.3, 280.96.1-56.bi.2.4, 280.96.1-56.bi.2.5, 280.96.1-56.bi.2.6, 280.96.1-56.bi.2.7, 280.96.1-56.bi.2.8, 280.96.1-56.bi.2.9, 280.96.1-56.bi.2.10, 280.96.1-56.bi.2.11, 280.96.1-56.bi.2.12, 280.96.1-56.bi.2.13, 280.96.1-56.bi.2.14, 280.96.1-56.bi.2.15, 280.96.1-56.bi.2.16
Cyclic 56-isogeny field degree:	$16$
Cyclic 56-torsion field degree:	$384$
Full 56-torsion field degree:	$64512$

Jacobian

Conductor:	$2^{6}\cdot7^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	3136.2.a.m

Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

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
8.24.0.e.1	$8$	$2$	$2$	$0$	$0$	full Jacobian
56.24.0.h.2	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.24.1.d.1	$56$	$2$	$2$	$1$	$1$	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
56.96.1.g.2	$56$	$2$	$2$	$1$	$1$	dimension zero
56.96.1.w.1	$56$	$2$	$2$	$1$	$1$	dimension zero
56.96.1.bl.1	$56$	$2$	$2$	$1$	$1$	dimension zero
56.96.1.bp.2	$56$	$2$	$2$	$1$	$1$	dimension zero
56.96.1.bu.1	$56$	$2$	$2$	$1$	$1$	dimension zero
56.96.1.by.2	$56$	$2$	$2$	$1$	$1$	dimension zero
56.96.1.cf.2	$56$	$2$	$2$	$1$	$1$	dimension zero
56.96.1.ch.1	$56$	$2$	$2$	$1$	$1$	dimension zero
56.384.25.eh.1	$56$	$8$	$8$	$25$	$4$	$1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.hd.2	$56$	$21$	$21$	$73$	$10$	$1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.hd.1	$56$	$28$	$28$	$97$	$13$	$1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
168.96.1.gm.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.gs.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.ht.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.hz.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.mu.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.na.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.oa.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.og.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.144.9.sh.2	$168$	$3$	$3$	$9$	$?$	not computed
168.192.9.jr.2	$168$	$4$	$4$	$9$	$?$	not computed
280.96.1.gm.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1.gs.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1.ht.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1.hz.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1.ma.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1.mg.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1.ng.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1.nm.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.240.17.ev.2	$280$	$5$	$5$	$17$	$?$	not computed
280.288.17.ly.1	$280$	$6$	$6$	$17$	$?$	not computed