Modular curve 56.1008.34-56.z.1.58

• Label: 56.1008.34-56.z.1.58
• Level: $56$
• Index: $1008$
• Genus: $34$
• Analytic rank: $1$
• Cusps: $18$
• $\Q$-cusps: $0$

Invariants

Level:	$56$		$\SL_2$-level:	$56$		Newform level:	$392$
Index:	$1008$		$\PSL_2$-index:	$504$
Genus:	$34 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$
Cusps:	$18$ (none of which are rational)		Cusp widths	$14^{6}\cdot28^{9}\cdot56^{3}$		Cusp orbits	$3^{4}\cdot6$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$5 \le \gamma \le 16$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 16$
Rational cusps:	$0$
Rational CM points:	none

Other labels

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

56.1008.34.9

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}1&16\\28&41\end{bmatrix}$, $\begin{bmatrix}5&48\\32&41\end{bmatrix}$, $\begin{bmatrix}21&24\\12&49\end{bmatrix}$, $\begin{bmatrix}35&32\\30&35\end{bmatrix}$, $\begin{bmatrix}43&28\\6&27\end{bmatrix}$, $\begin{bmatrix}53&28\\10&3\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.504.34.z.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree:	$16$
Cyclic 56-torsion field degree:	$192$
Full 56-torsion field degree:	$3072$

Jacobian

Conductor:	$2^{84}\cdot7^{68}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{4}\cdot2^{6}\cdot6\cdot12$
Newforms:	98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 392.2.a.c, 392.2.a.f, 392.2.a.g, 392.2.b.e, 392.2.b.g

Rational points

This modular curve has no $\Q_p$ points for $p=3,11,67$, and therefore no 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
$X_{\mathrm{ns}}^+(7)$	$7$	$48$	$24$	$0$	$0$	full Jacobian
8.48.0-8.e.2.13	$8$	$21$	$21$	$0$	$0$	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.e.2.13	$8$	$21$	$21$	$0$	$0$	full Jacobian
28.504.16-28.b.1.11	$28$	$2$	$2$	$16$	$1$	$6\cdot12$
56.504.16-28.b.1.1	$56$	$2$	$2$	$16$	$1$	$6\cdot12$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.2016.67-56.en.1.26	$56$	$2$	$2$	$67$	$5$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.er.1.26	$56$	$2$	$2$	$67$	$16$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.ev.2.28	$56$	$2$	$2$	$67$	$10$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.ez.2.26	$56$	$2$	$2$	$67$	$1$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.ff.2.44	$56$	$2$	$2$	$67$	$2$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.fj.1.26	$56$	$2$	$2$	$67$	$8$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.fn.2.25	$56$	$2$	$2$	$67$	$8$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.fr.2.26	$56$	$2$	$2$	$67$	$6$	$1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.70-56.h.1.7	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.l.1.1	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.t.1.7	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.x.1.7	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bd.1.5	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bf.1.8	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bl.1.7	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bn.1.7	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bt.1.26	$56$	$2$	$2$	$70$	$7$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.bv.1.26	$56$	$2$	$2$	$70$	$7$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cb.1.26	$56$	$2$	$2$	$70$	$10$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cd.1.26	$56$	$2$	$2$	$70$	$10$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cj.1.25	$56$	$2$	$2$	$70$	$6$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cl.1.25	$56$	$2$	$2$	$70$	$6$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cr.1.26	$56$	$2$	$2$	$70$	$9$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.ct.1.26	$56$	$2$	$2$	$70$	$9$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cz.2.14	$56$	$2$	$2$	$70$	$9$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.db.2.15	$56$	$2$	$2$	$70$	$9$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dh.2.13	$56$	$2$	$2$	$70$	$6$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dj.2.16	$56$	$2$	$2$	$70$	$6$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dp.2.14	$56$	$2$	$2$	$70$	$10$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dr.2.12	$56$	$2$	$2$	$70$	$10$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dx.2.15	$56$	$2$	$2$	$70$	$7$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dz.2.9	$56$	$2$	$2$	$70$	$7$	$1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.eh.1.26	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.ej.1.26	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.ep.1.25	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.er.1.25	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.ez.1.26	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.fd.1.26	$56$	$2$	$2$	$70$	$10$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.fl.1.26	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.fp.1.22	$56$	$2$	$2$	$70$	$6$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.73-56.df.1.25	$56$	$2$	$2$	$73$	$9$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.dn.1.21	$56$	$2$	$2$	$73$	$9$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.ee.1.26	$56$	$2$	$2$	$73$	$9$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.ei.1.26	$56$	$2$	$2$	$73$	$9$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.gv.1.26	$56$	$2$	$2$	$73$	$10$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.gx.1.26	$56$	$2$	$2$	$73$	$10$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.hd.1.26	$56$	$2$	$2$	$73$	$10$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.hf.1.26	$56$	$2$	$2$	$73$	$10$	$1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.jt.1.26	$56$	$2$	$2$	$73$	$12$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.jv.1.26	$56$	$2$	$2$	$73$	$12$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kb.1.26	$56$	$2$	$2$	$73$	$8$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kd.1.26	$56$	$2$	$2$	$73$	$8$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kj.1.26	$56$	$2$	$2$	$73$	$12$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kl.1.26	$56$	$2$	$2$	$73$	$12$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kr.1.25	$56$	$2$	$2$	$73$	$10$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kt.1.25	$56$	$2$	$2$	$73$	$10$	$1^{9}\cdot2^{8}\cdot4^{2}\cdot6$