Modular curve 56.96.1.ck.1

• Label: 56.96.1.ck.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	$2^{4}\cdot8$
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:	56.96.1.1126

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}13&16\\40&29\end{bmatrix}$, $\begin{bmatrix}14&9\\5&26\end{bmatrix}$, $\begin{bmatrix}33&2\\34&17\end{bmatrix}$, $\begin{bmatrix}55&28\\52&23\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.192.1-56.ck.1.1, 56.192.1-56.ck.1.2, 56.192.1-56.ck.1.3, 56.192.1-56.ck.1.4, 56.192.1-56.ck.1.5, 56.192.1-56.ck.1.6, 56.192.1-56.ck.1.7, 56.192.1-56.ck.1.8, 112.192.1-56.ck.1.1, 112.192.1-56.ck.1.2, 112.192.1-56.ck.1.3, 112.192.1-56.ck.1.4, 112.192.1-56.ck.1.5, 112.192.1-56.ck.1.6, 112.192.1-56.ck.1.7, 112.192.1-56.ck.1.8, 112.192.1-56.ck.1.9, 112.192.1-56.ck.1.10, 112.192.1-56.ck.1.11, 112.192.1-56.ck.1.12, 112.192.1-56.ck.1.13, 112.192.1-56.ck.1.14, 112.192.1-56.ck.1.15, 112.192.1-56.ck.1.16, 168.192.1-56.ck.1.1, 168.192.1-56.ck.1.2, 168.192.1-56.ck.1.3, 168.192.1-56.ck.1.4, 168.192.1-56.ck.1.5, 168.192.1-56.ck.1.6, 168.192.1-56.ck.1.7, 168.192.1-56.ck.1.8, 280.192.1-56.ck.1.1, 280.192.1-56.ck.1.2, 280.192.1-56.ck.1.3, 280.192.1-56.ck.1.4, 280.192.1-56.ck.1.5, 280.192.1-56.ck.1.6, 280.192.1-56.ck.1.7, 280.192.1-56.ck.1.8
Cyclic 56-isogeny field degree:	$8$
Cyclic 56-torsion field degree:	$192$
Full 56-torsion field degree:	$32256$

Jacobian

Conductor:	$2^{6}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	64.2.a.a

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

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
8.48.1.bc.1	$8$	$2$	$2$	$1$	$0$	dimension zero
56.48.0.bd.1	$56$	$2$	$2$	$0$	$0$	full Jacobian
56.48.0.bd.2	$56$	$2$	$2$	$0$	$0$	full Jacobian

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.ro.1	$56$	$8$	$8$	$49$	$6$	$1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.cku.1	$56$	$21$	$21$	$145$	$23$	$1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.cjq.1	$56$	$28$	$28$	$193$	$29$	$1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
112.192.5.gn.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.go.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.gx.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.he.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.hf.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.hf.2	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.hg.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.hg.2	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.hk.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.hr.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.ht.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.5.hu.1	$112$	$2$	$2$	$5$	$?$	not computed
112.192.9.ma.1	$112$	$2$	$2$	$9$	$?$	not computed
112.192.9.ma.2	$112$	$2$	$2$	$9$	$?$	not computed
112.192.9.mb.1	$112$	$2$	$2$	$9$	$?$	not computed
112.192.9.mb.2	$112$	$2$	$2$	$9$	$?$	not computed
168.288.17.osr.1	$168$	$3$	$3$	$17$	$?$	not computed
168.384.17.eqn.1	$168$	$4$	$4$	$17$	$?$	not computed