Modular curve 56.96.1.x.1

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

Invariants

Level:	$56$		$\SL_2$-level:	$8$		Newform level:	$1568$
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	$2^{6}\cdot4$
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:	8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	56.96.1.178

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}3&4\\0&51\end{bmatrix}$, $\begin{bmatrix}3&4\\38&33\end{bmatrix}$, $\begin{bmatrix}7&48\\18&9\end{bmatrix}$, $\begin{bmatrix}43&12\\26&9\end{bmatrix}$, $\begin{bmatrix}47&24\\46&21\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.192.1-56.x.1.1, 56.192.1-56.x.1.2, 56.192.1-56.x.1.3, 56.192.1-56.x.1.4, 56.192.1-56.x.1.5, 56.192.1-56.x.1.6, 56.192.1-56.x.1.7, 56.192.1-56.x.1.8, 56.192.1-56.x.1.9, 56.192.1-56.x.1.10, 56.192.1-56.x.1.11, 56.192.1-56.x.1.12, 112.192.1-56.x.1.1, 112.192.1-56.x.1.2, 112.192.1-56.x.1.3, 112.192.1-56.x.1.4, 112.192.1-56.x.1.5, 112.192.1-56.x.1.6, 112.192.1-56.x.1.7, 112.192.1-56.x.1.8, 168.192.1-56.x.1.1, 168.192.1-56.x.1.2, 168.192.1-56.x.1.3, 168.192.1-56.x.1.4, 168.192.1-56.x.1.5, 168.192.1-56.x.1.6, 168.192.1-56.x.1.7, 168.192.1-56.x.1.8, 168.192.1-56.x.1.9, 168.192.1-56.x.1.10, 168.192.1-56.x.1.11, 168.192.1-56.x.1.12, 280.192.1-56.x.1.1, 280.192.1-56.x.1.2, 280.192.1-56.x.1.3, 280.192.1-56.x.1.4, 280.192.1-56.x.1.5, 280.192.1-56.x.1.6, 280.192.1-56.x.1.7, 280.192.1-56.x.1.8, 280.192.1-56.x.1.9, 280.192.1-56.x.1.10, 280.192.1-56.x.1.11, 280.192.1-56.x.1.12
Cyclic 56-isogeny field degree:	$16$
Cyclic 56-torsion field degree:	$192$
Full 56-torsion field degree:	$32256$

Jacobian

Conductor:	$2^{5}\cdot7^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	1568.2.a.e

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
8.48.0.c.1	$8$	$2$	$2$	$0$	$0$	full Jacobian
56.48.0.b.2	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.0.u.1	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.0.v.1	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.1.o.2	$56$	$2$	$2$	$1$	$1$	dimension zero
56.48.1.be.2	$56$	$2$	$2$	$1$	$1$	dimension zero
56.48.1.bf.2	$56$	$2$	$2$	$1$	$1$	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
56.192.5.x.1	$56$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
56.192.5.y.2	$56$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
56.192.5.ba.2	$56$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
56.192.5.bb.3	$56$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
56.768.49.fv.2	$56$	$8$	$8$	$49$	$7$	$1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.pj.2	$56$	$21$	$21$	$145$	$20$	$1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.qd.2	$56$	$28$	$28$	$193$	$26$	$1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
112.192.5.a.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.g.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.p.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.r.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.cq.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.cs.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.db.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.dh.1	$112$	$2$	$2$	$5$	$?$	not computed
168.192.5.hi.2	$168$	$2$	$2$	$5$	$?$	not computed
168.192.5.hk.2	$168$	$2$	$2$	$5$	$?$	not computed
168.192.5.hs.2	$168$	$2$	$2$	$5$	$?$	not computed
168.192.5.hu.1	$168$	$2$	$2$	$5$	$?$	not computed
168.288.17.bwl.2	$168$	$3$	$3$	$17$	$?$	not computed
168.384.17.tn.2	$168$	$4$	$4$	$17$	$?$	not computed
280.192.5.ha.2	$280$	$2$	$2$	$5$	$?$	not computed
280.192.5.hc.2	$280$	$2$	$2$	$5$	$?$	not computed
280.192.5.hk.2	$280$	$2$	$2$	$5$	$?$	not computed
280.192.5.hm.2	$280$	$2$	$2$	$5$	$?$	not computed