Modular curve 56.1344.45-56.u.1.62

• Label: 56.1344.45-56.u.1.62
• Level: $56$
• Index: $1344$
• Genus: $45$
• Analytic rank: $1$
• Cusps: $24$
• $\Q$-cusps: $2$

Invariants

Level:	$56$		$\SL_2$-level:	$56$		Newform level:	$1568$
Index:	$1344$		$\PSL_2$-index:	$672$
Genus:	$45 = 1 + \frac{ 672 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps:	$24$ (of which $2$ are rational)		Cusp widths	$14^{8}\cdot28^{12}\cdot56^{4}$		Cusp orbits	$1^{2}\cdot2^{2}\cdot3^{2}\cdot6^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$7 \le \gamma \le 24$
$\overline{\Q}$-gonality:	$7 \le \gamma \le 24$
Rational cusps:	$2$
Rational CM points:	none

Other labels

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

56.1344.45.95

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}17&8\\48&25\end{bmatrix}$, $\begin{bmatrix}37&52\\38&33\end{bmatrix}$, $\begin{bmatrix}39&0\\28&25\end{bmatrix}$, $\begin{bmatrix}45&8\\52&39\end{bmatrix}$, $\begin{bmatrix}45&36\\30&25\end{bmatrix}$, $\begin{bmatrix}51&28\\40&13\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.672.45.u.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree:	$4$
Cyclic 56-torsion field degree:	$96$
Full 56-torsion field degree:	$2304$

Jacobian

Conductor:	$2^{159}\cdot7^{79}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{9}\cdot2^{7}\cdot4\cdot6\cdot12$
Newforms:	14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 224.2.b.a, 224.2.b.b, 392.2.a.c, 392.2.a.f, 392.2.a.g, 1568.2.b.f, 1568.2.b.g

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points.

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{sp}}^+(7)$	$7$	$48$	$24$	$0$	$0$	full Jacobian
8.48.0-8.d.1.16	$8$	$28$	$28$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.48.0-8.d.1.16	$8$	$28$	$28$	$0$	$0$	full Jacobian
56.672.21-28.b.1.1	$56$	$2$	$2$	$21$	$1$	$2\cdot4\cdot6\cdot12$
56.672.21-28.b.1.24	$56$	$2$	$2$	$21$	$1$	$2\cdot4\cdot6\cdot12$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.2688.89-56.ef.1.24	$56$	$2$	$2$	$89$	$5$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.89-56.ej.2.31	$56$	$2$	$2$	$89$	$17$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.89-56.eo.1.30	$56$	$2$	$2$	$89$	$11$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.89-56.es.1.29	$56$	$2$	$2$	$89$	$2$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.89-56.ex.1.22	$56$	$2$	$2$	$89$	$3$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.89-56.fb.1.28	$56$	$2$	$2$	$89$	$13$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.89-56.fg.1.28	$56$	$2$	$2$	$89$	$11$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.89-56.fk.1.26	$56$	$2$	$2$	$89$	$6$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.2688.93-56.a.1.6	$56$	$2$	$2$	$93$	$7$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.h.1.4	$56$	$2$	$2$	$93$	$7$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.m.1.1	$56$	$2$	$2$	$93$	$12$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.t.2.4	$56$	$2$	$2$	$93$	$12$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.y.2.11	$56$	$2$	$2$	$93$	$8$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.ba.1.10	$56$	$2$	$2$	$93$	$8$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.bg.1.9	$56$	$2$	$2$	$93$	$13$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.bi.2.12	$56$	$2$	$2$	$93$	$13$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.bo.1.30	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.bq.1.30	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.bw.1.31	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.by.1.30	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.ce.1.28	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.cg.1.28	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.cm.1.30	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.co.1.28	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.cu.1.25	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.cw.1.28	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.dc.1.27	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.de.1.26	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.dk.1.18	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.dm.1.24	$56$	$2$	$2$	$93$	$12$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.ds.1.22	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.du.1.20	$56$	$2$	$2$	$93$	$8$	$1^{20}\cdot2^{5}\cdot4^{3}\cdot6$
56.2688.93-56.ec.1.23	$56$	$2$	$2$	$93$	$13$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.ee.1.22	$56$	$2$	$2$	$93$	$13$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.ek.1.28	$56$	$2$	$2$	$93$	$8$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.em.1.28	$56$	$2$	$2$	$93$	$8$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.es.1.31	$56$	$2$	$2$	$93$	$12$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.ez.1.30	$56$	$2$	$2$	$93$	$12$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.fe.1.24	$56$	$2$	$2$	$93$	$7$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.93-56.fl.1.30	$56$	$2$	$2$	$93$	$7$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.2688.97-56.cm.1.24	$56$	$2$	$2$	$97$	$10$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.df.1.28	$56$	$2$	$2$	$97$	$10$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.dx.1.23	$56$	$2$	$2$	$97$	$12$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.ee.1.22	$56$	$2$	$2$	$97$	$12$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.gq.1.30	$56$	$2$	$2$	$97$	$13$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.gs.1.30	$56$	$2$	$2$	$97$	$13$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.gy.1.31	$56$	$2$	$2$	$97$	$13$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.ha.1.30	$56$	$2$	$2$	$97$	$13$	$1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$
56.2688.97-56.jo.1.31	$56$	$2$	$2$	$97$	$15$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.2688.97-56.jq.1.30	$56$	$2$	$2$	$97$	$15$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.2688.97-56.jw.1.30	$56$	$2$	$2$	$97$	$11$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.2688.97-56.jy.1.30	$56$	$2$	$2$	$97$	$11$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.2688.97-56.ke.1.30	$56$	$2$	$2$	$97$	$15$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.2688.97-56.kg.1.28	$56$	$2$	$2$	$97$	$15$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.2688.97-56.km.1.28	$56$	$2$	$2$	$97$	$11$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.2688.97-56.ko.1.28	$56$	$2$	$2$	$97$	$11$	$1^{12}\cdot2^{11}\cdot4^{3}\cdot6$
56.4032.133-56.cr.1.42	$56$	$3$	$3$	$133$	$7$	$1^{26}\cdot2^{10}\cdot4^{3}\cdot6^{3}\cdot12$