Modular curve 56.2016.145.pj.1

• Label: 56.2016.145.pj.1
• Level: $56$
• Index: $2016$
• Genus: $145$
• Analytic rank: $20$
• Cusps: $48$
• $\Q$-cusps: $0$

Invariants

Level:	$56$		$\SL_2$-level:	$56$		Newform level:	$1568$
Index:	$2016$		$\PSL_2$-index:	$2016$
Genus:	$145 = 1 + \frac{ 2016 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$
Cusps:	$48$ (none of which are rational)		Cusp widths	$28^{24}\cdot56^{24}$		Cusp orbits	$3^{4}\cdot12^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$20$
$\Q$-gonality:	$20 \le \gamma \le 42$
$\overline{\Q}$-gonality:	$20 \le \gamma \le 42$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label:

56.2016.145.831

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}13&0\\24&1\end{bmatrix}$, $\begin{bmatrix}15&34\\8&25\end{bmatrix}$, $\begin{bmatrix}23&4\\24&39\end{bmatrix}$, $\begin{bmatrix}23&20\\28&19\end{bmatrix}$, $\begin{bmatrix}49&34\\36&3\end{bmatrix}$, $\begin{bmatrix}55&14\\28&13\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.4032.145-56.pj.1.1, 56.4032.145-56.pj.1.2, 56.4032.145-56.pj.1.3, 56.4032.145-56.pj.1.4, 56.4032.145-56.pj.1.5, 56.4032.145-56.pj.1.6, 56.4032.145-56.pj.1.7, 56.4032.145-56.pj.1.8, 56.4032.145-56.pj.1.9, 56.4032.145-56.pj.1.10, 56.4032.145-56.pj.1.11, 56.4032.145-56.pj.1.12, 56.4032.145-56.pj.1.13, 56.4032.145-56.pj.1.14, 56.4032.145-56.pj.1.15, 56.4032.145-56.pj.1.16, 56.4032.145-56.pj.1.17, 56.4032.145-56.pj.1.18, 56.4032.145-56.pj.1.19, 56.4032.145-56.pj.1.20, 56.4032.145-56.pj.1.21, 56.4032.145-56.pj.1.22, 56.4032.145-56.pj.1.23, 56.4032.145-56.pj.1.24
Cyclic 56-isogeny field degree:	$16$
Cyclic 56-torsion field degree:	$192$
Full 56-torsion field degree:	$1536$

Jacobian

Conductor:	$2^{499}\cdot7^{290}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{17}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
Newforms:	98.2.a.b$^{5}$, 196.2.a.a, 196.2.a.b$^{3}$, 196.2.a.c$^{4}$, 392.2.a.a, 392.2.a.c$^{2}$, 392.2.a.e, 392.2.a.f$^{2}$, 392.2.a.g$^{3}$, 392.2.b.e$^{2}$, 392.2.b.f, 392.2.b.g$^{3}$, 784.2.a.a, 784.2.a.c, 784.2.a.d, 784.2.a.g, 784.2.a.h, 784.2.a.j, 784.2.a.k$^{2}$, 784.2.a.l$^{2}$, 784.2.a.m$^{2}$, 1568.2.a.e, 1568.2.a.j, 1568.2.a.n, 1568.2.a.o, 1568.2.a.p, 1568.2.a.q, 1568.2.a.r, 1568.2.a.t, 1568.2.a.u, 1568.2.a.x, 1568.2.b.e, 1568.2.b.g

Rational points

This modular curve has no $\Q_p$ points for $p=3,5,11,\ldots,373$, and therefore no rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.96.1.x.2	$56$	$21$	$21$	$1$	$1$	$1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.1008.70.g.1	$56$	$2$	$2$	$70$	$6$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1008.70.l.1	$56$	$2$	$2$	$70$	$6$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1008.70.ep.1	$56$	$2$	$2$	$70$	$6$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1008.70.er.1	$56$	$2$	$2$	$70$	$6$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1008.73.cu.1	$56$	$2$	$2$	$73$	$20$	$6^{4}\cdot12^{4}$
56.1008.73.gv.1	$56$	$2$	$2$	$73$	$10$	$1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$
56.1008.73.gx.1	$56$	$2$	$2$	$73$	$10$	$1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.4032.289.gt.1	$56$	$2$	$2$	$289$	$42$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.289.hn.1	$56$	$2$	$2$	$289$	$63$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.289.mx.2	$56$	$2$	$2$	$289$	$41$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.289.nt.2	$56$	$2$	$2$	$289$	$55$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.289.wf.2	$56$	$2$	$2$	$289$	$53$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.289.wz.2	$56$	$2$	$2$	$289$	$47$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.289.bak.1	$56$	$2$	$2$	$289$	$48$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.289.bbf.2	$56$	$2$	$2$	$289$	$47$	$1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.4032.301.ri.1	$56$	$2$	$2$	$301$	$51$	$1^{14}\cdot2^{29}\cdot4^{5}\cdot6^{2}\cdot12^{3}\cdot16$
56.4032.301.rk.2	$56$	$2$	$2$	$301$	$51$	$1^{14}\cdot2^{29}\cdot4^{5}\cdot6^{2}\cdot12^{3}\cdot16$
56.4032.301.rq.2	$56$	$2$	$2$	$301$	$51$	$1^{14}\cdot2^{29}\cdot4^{5}\cdot6^{2}\cdot12^{3}\cdot16$
56.4032.301.rt.2	$56$	$2$	$2$	$301$	$51$	$1^{14}\cdot2^{29}\cdot4^{5}\cdot6^{2}\cdot12^{3}\cdot16$
56.4032.301.ry.2	$56$	$2$	$2$	$301$	$54$	$1^{46}\cdot2^{23}\cdot4^{8}\cdot6^{2}\cdot8\cdot12$
56.4032.301.rz.2	$56$	$2$	$2$	$301$	$52$	$1^{46}\cdot2^{23}\cdot4^{8}\cdot6^{2}\cdot8\cdot12$
56.4032.301.sf.2	$56$	$2$	$2$	$301$	$54$	$1^{46}\cdot2^{23}\cdot4^{8}\cdot6^{2}\cdot8\cdot12$
56.4032.301.sg.2	$56$	$2$	$2$	$301$	$52$	$1^{46}\cdot2^{23}\cdot4^{8}\cdot6^{2}\cdot8\cdot12$