Modular curve 56.2016.145.pg.2

• Label: 56.2016.145.pg.2
• Level: $56$
• Index: $2016$
• Genus: $145$
• Analytic rank: $18$
• 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	$3^{4}\cdot6^{4}\cdot12$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$18$
$\Q$-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:

56.2016.145.4664

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}7&20\\22&21\end{bmatrix}$, $\begin{bmatrix}19&24\\10&49\end{bmatrix}$, $\begin{bmatrix}21&8\\6&7\end{bmatrix}$, $\begin{bmatrix}29&48\\4&41\end{bmatrix}$, $\begin{bmatrix}47&40\\54&41\end{bmatrix}$, $\begin{bmatrix}55&44\\44&43\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.4032.145-56.pg.2.1, 56.4032.145-56.pg.2.2, 56.4032.145-56.pg.2.3, 56.4032.145-56.pg.2.4, 56.4032.145-56.pg.2.5, 56.4032.145-56.pg.2.6, 56.4032.145-56.pg.2.7, 56.4032.145-56.pg.2.8, 56.4032.145-56.pg.2.9, 56.4032.145-56.pg.2.10, 56.4032.145-56.pg.2.11, 56.4032.145-56.pg.2.12, 56.4032.145-56.pg.2.13, 56.4032.145-56.pg.2.14, 56.4032.145-56.pg.2.15, 56.4032.145-56.pg.2.16, 56.4032.145-56.pg.2.17, 56.4032.145-56.pg.2.18, 56.4032.145-56.pg.2.19, 56.4032.145-56.pg.2.20, 56.4032.145-56.pg.2.21, 56.4032.145-56.pg.2.22, 56.4032.145-56.pg.2.23, 56.4032.145-56.pg.2.24
Cyclic 56-isogeny field degree:	$8$
Cyclic 56-torsion field degree:	$192$
Full 56-torsion field degree:	$1536$

Jacobian

Conductor:	$2^{500}\cdot7^{264}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{33}\cdot2^{24}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
Newforms:	14.2.a.a, 32.2.a.a, 49.2.a.a, 56.2.a.a, 56.2.a.b, 56.2.b.a, 56.2.b.b, 98.2.a.b$^{4}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b$^{4}$, 196.2.a.c$^{3}$, 224.2.a.a, 224.2.a.b, 224.2.a.c, 224.2.a.d, 224.2.b.a, 224.2.b.b, 392.2.a.c$^{3}$, 392.2.a.f$^{3}$, 392.2.a.g$^{2}$, 392.2.a.h, 392.2.b.a, 392.2.b.d, 392.2.b.e$^{3}$, 392.2.b.g$^{2}$, 784.2.a.a$^{2}$, 784.2.a.d$^{2}$, 784.2.a.f, 784.2.a.h$^{2}$, 784.2.a.k, 784.2.a.l, 784.2.a.m, 784.2.a.n, 1568.2.a.a, 1568.2.a.c, 1568.2.a.d, 1568.2.a.f, 1568.2.a.g, 1568.2.a.i, 1568.2.a.l, 1568.2.a.m, 1568.2.a.s, 1568.2.a.w, 1568.2.b.b, 1568.2.b.c, 1568.2.b.f

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
56.1008.70.f.2	$56$	$2$	$2$	$70$	$6$	$1^{23}\cdot2^{8}\cdot6^{2}\cdot12^{2}$
56.1008.70.l.1	$56$	$2$	$2$	$70$	$6$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.1008.70.cj.2	$56$	$2$	$2$	$70$	$6$	$1^{15}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12$
56.1008.70.cl.2	$56$	$2$	$2$	$70$	$6$	$1^{15}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12$
56.1008.73.cv.1	$56$	$2$	$2$	$73$	$18$	$2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
56.1008.73.kb.2	$56$	$2$	$2$	$73$	$8$	$1^{20}\cdot2^{10}\cdot4^{2}\cdot6^{2}\cdot12$
56.1008.73.kd.2	$56$	$2$	$2$	$73$	$8$	$1^{20}\cdot2^{10}\cdot4^{2}\cdot6^{2}\cdot12$

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.2	$56$	$2$	$2$	$289$	$42$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.hm.1	$56$	$2$	$2$	$289$	$60$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.my.1	$56$	$2$	$2$	$289$	$37$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.nr.1	$56$	$2$	$2$	$289$	$50$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.wd.1	$56$	$2$	$2$	$289$	$48$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.xa.1	$56$	$2$	$2$	$289$	$43$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.baj.1	$56$	$2$	$2$	$289$	$45$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.bbf.1	$56$	$2$	$2$	$289$	$47$	$1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.301.rf.1	$56$	$2$	$2$	$301$	$47$	$1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.rl.1	$56$	$2$	$2$	$301$	$50$	$1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.ro.1	$56$	$2$	$2$	$301$	$51$	$1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.rs.1	$56$	$2$	$2$	$301$	$52$	$1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.ru.2	$56$	$2$	$2$	$301$	$47$	$1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$
56.4032.301.sa.3	$56$	$2$	$2$	$301$	$50$	$1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$
56.4032.301.sc.2	$56$	$2$	$2$	$301$	$51$	$1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$
56.4032.301.sg.2	$56$	$2$	$2$	$301$	$52$	$1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$