Modular curve 56.192.5.bb.4

• Label: 56.192.5.bb.4
• Level: $56$
• Index: $192$
• Genus: $5$
• Analytic rank: $2$
• Cusps: $24$
• $\Q$-cusps: $4$

Invariants

Level:	$56$		$\SL_2$-level:	$8$		Newform level:	$3136$
Index:	$192$		$\PSL_2$-index:	$192$
Genus:	$5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps:	$24$ (of which $4$ are rational)		Cusp widths	$8^{24}$		Cusp orbits	$1^{4}\cdot2^{2}\cdot4^{2}\cdot8$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$2$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$4$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	8A5
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	56.192.5.103

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}11&12\\28&33\end{bmatrix}$, $\begin{bmatrix}13&12\\20&7\end{bmatrix}$, $\begin{bmatrix}41&40\\20&31\end{bmatrix}$, $\begin{bmatrix}49&4\\8&33\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.384.5-56.bb.4.1, 56.384.5-56.bb.4.2, 56.384.5-56.bb.4.3, 56.384.5-56.bb.4.4, 56.384.5-56.bb.4.5, 56.384.5-56.bb.4.6, 56.384.5-56.bb.4.7, 56.384.5-56.bb.4.8, 112.384.5-56.bb.4.1, 112.384.5-56.bb.4.2, 112.384.5-56.bb.4.3, 112.384.5-56.bb.4.4, 112.384.5-56.bb.4.5, 112.384.5-56.bb.4.6, 112.384.5-56.bb.4.7, 112.384.5-56.bb.4.8, 112.384.5-56.bb.4.9, 112.384.5-56.bb.4.10, 112.384.5-56.bb.4.11, 112.384.5-56.bb.4.12, 112.384.5-56.bb.4.13, 112.384.5-56.bb.4.14, 112.384.5-56.bb.4.15, 112.384.5-56.bb.4.16, 168.384.5-56.bb.4.1, 168.384.5-56.bb.4.2, 168.384.5-56.bb.4.3, 168.384.5-56.bb.4.4, 168.384.5-56.bb.4.5, 168.384.5-56.bb.4.6, 168.384.5-56.bb.4.7, 168.384.5-56.bb.4.8, 280.384.5-56.bb.4.1, 280.384.5-56.bb.4.2, 280.384.5-56.bb.4.3, 280.384.5-56.bb.4.4, 280.384.5-56.bb.4.5, 280.384.5-56.bb.4.6, 280.384.5-56.bb.4.7, 280.384.5-56.bb.4.8
Cyclic 56-isogeny field degree:	$8$
Cyclic 56-torsion field degree:	$192$
Full 56-torsion field degree:	$16128$

Jacobian

Conductor:	$2^{28}\cdot7^{8}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	32.2.a.a, 1568.2.a.e, 3136.2.a.m, 3136.2.b.b

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal 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
8.96.1.g.1	$8$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
56.96.1.n.2	$56$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
56.96.1.x.2	$56$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
56.96.3.o.3	$56$	$2$	$2$	$3$	$2$	$2$
56.96.3.p.2	$56$	$2$	$2$	$3$	$0$	$1^{2}$
56.96.3.r.2	$56$	$2$	$2$	$3$	$1$	$1^{2}$
56.96.3.w.1	$56$	$2$	$2$	$3$	$1$	$1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.1536.105.gl.3	$56$	$8$	$8$	$105$	$16$	$1^{38}\cdot2^{15}\cdot4^{8}$
56.4032.301.rt.2	$56$	$21$	$21$	$301$	$51$	$1^{28}\cdot2^{54}\cdot4^{6}\cdot6^{6}\cdot12^{7}\cdot16$
56.5376.401.th.1	$56$	$28$	$28$	$401$	$65$	$1^{66}\cdot2^{69}\cdot4^{14}\cdot6^{6}\cdot12^{7}\cdot16$
112.384.13.d.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.q.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.bn.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.ca.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.da.3	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.da.4	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.db.3	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.db.4	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.ds.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.eh.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.fe.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.13.ft.2	$112$	$2$	$2$	$13$	$?$	not computed
112.384.17.dt.1	$112$	$2$	$2$	$17$	$?$	not computed
112.384.17.dt.3	$112$	$2$	$2$	$17$	$?$	not computed
112.384.17.ea.1	$112$	$2$	$2$	$17$	$?$	not computed
112.384.17.ea.3	$112$	$2$	$2$	$17$	$?$	not computed