Modular curve 56.2688.89-56.fa.2.32

• Label: 56.2688.89-56.fa.2.32
• Level: $56$
• Index: $2688$
• Genus: $89$
• Analytic rank: $3$
• Cusps: $48$
• $\Q$-cusps: $0$

Invariants

Level:	$56$		$\SL_2$-level:	$56$		Newform level:	$392$
Index:	$2688$		$\PSL_2$-index:	$1344$
Genus:	$89 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$
Cusps:	$48$ (none of which are rational)		Cusp widths	$14^{16}\cdot28^{24}\cdot56^{8}$		Cusp orbits	$2^{4}\cdot3^{8}\cdot4\cdot6^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$3$
$\Q$-gonality:	$14 \le \gamma \le 48$
$\overline{\Q}$-gonality:	$14 \le \gamma \le 48$
Rational cusps:	$0$
Rational CM points:	none

Other labels

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

56.2688.89.10280

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}33&28\\14&47\end{bmatrix}$, $\begin{bmatrix}33&28\\42&19\end{bmatrix}$, $\begin{bmatrix}33&48\\0&9\end{bmatrix}$, $\begin{bmatrix}33&48\\28&51\end{bmatrix}$, $\begin{bmatrix}39&36\\44&31\end{bmatrix}$, $\begin{bmatrix}55&28\\0&55\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.1344.89.fa.2 for the level structure with $-I$)
Cyclic 56-isogeny field degree:	$4$
Cyclic 56-torsion field degree:	$96$
Full 56-torsion field degree:	$1152$

Jacobian

Conductor:	$2^{217}\cdot7^{156}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{27}\cdot2^{11}\cdot4^{4}\cdot6^{2}\cdot12$
Newforms:	14.2.a.a$^{6}$, 49.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 98.2.a.a$^{3}$, 98.2.a.b$^{3}$, 196.2.a.b$^{4}$, 196.2.a.c$^{2}$, 392.2.a.b, 392.2.a.c$^{2}$, 392.2.a.d, 392.2.a.f$^{2}$, 392.2.a.g, 392.2.a.h, 392.2.b.a, 392.2.b.b, 392.2.b.c, 392.2.b.d, 392.2.b.e$^{2}$, 392.2.b.g

Rational points

This modular curve has no $\Q_p$ points for $p=3,5,17,19,59,131$, 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
7.56.1.b.1	$7$	$48$	$24$	$1$	$0$	$1^{26}\cdot2^{11}\cdot4^{4}\cdot6^{2}\cdot12$
8.48.0-8.e.1.7	$8$	$56$	$56$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.1344.41-28.j.1.11	$56$	$2$	$2$	$41$	$3$	$2^{4}\cdot4^{4}\cdot6^{2}\cdot12$
56.1344.41-28.j.1.33	$56$	$2$	$2$	$41$	$3$	$2^{4}\cdot4^{4}\cdot6^{2}\cdot12$
56.1344.45-56.x.2.40	$56$	$2$	$2$	$45$	$1$	$1^{18}\cdot2^{2}\cdot4\cdot6\cdot12$
56.1344.45-56.x.2.43	$56$	$2$	$2$	$45$	$1$	$1^{18}\cdot2^{2}\cdot4\cdot6\cdot12$
56.1344.45-56.bb.1.33	$56$	$2$	$2$	$45$	$1$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$
56.1344.45-56.bb.1.48	$56$	$2$	$2$	$45$	$1$	$1^{18}\cdot2^{4}\cdot4^{3}\cdot6$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.5376.177-56.cr.2.24	$56$	$2$	$2$	$177$	$3$	$2^{6}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.177-56.cv.2.24	$56$	$2$	$2$	$177$	$3$	$2^{6}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.177-56.cz.2.24	$56$	$2$	$2$	$177$	$3$	$2^{6}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.177-56.dd.2.24	$56$	$2$	$2$	$177$	$3$	$2^{6}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.bk.2.19	$56$	$2$	$2$	$185$	$16$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.bm.1.9	$56$	$2$	$2$	$185$	$16$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.bv.2.15	$56$	$2$	$2$	$185$	$25$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.bx.1.16	$56$	$2$	$2$	$185$	$25$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.gg.2.20	$56$	$2$	$2$	$185$	$17$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.gh.1.10	$56$	$2$	$2$	$185$	$17$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.gl.1.10	$56$	$2$	$2$	$185$	$26$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.gm.2.20	$56$	$2$	$2$	$185$	$26$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.id.2.28	$56$	$2$	$2$	$185$	$5$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.ih.2.24	$56$	$2$	$2$	$185$	$7$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.il.2.28	$56$	$2$	$2$	$185$	$5$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.ip.2.24	$56$	$2$	$2$	$185$	$7$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.jj.2.23	$56$	$2$	$2$	$185$	$7$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.jn.2.23	$56$	$2$	$2$	$185$	$5$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.jr.2.24	$56$	$2$	$2$	$185$	$7$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.jv.2.24	$56$	$2$	$2$	$185$	$5$	$2^{10}\cdot4^{8}\cdot8^{4}\cdot12$
56.5376.185-56.kz.2.23	$56$	$2$	$2$	$185$	$26$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.la.1.24	$56$	$2$	$2$	$185$	$26$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.le.2.24	$56$	$2$	$2$	$185$	$17$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.lf.2.23	$56$	$2$	$2$	$185$	$17$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.pe.2.21	$56$	$2$	$2$	$185$	$25$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.pg.1.23	$56$	$2$	$2$	$185$	$25$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.pp.2.22	$56$	$2$	$2$	$185$	$16$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.185-56.pr.2.14	$56$	$2$	$2$	$185$	$16$	$1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$
56.5376.193-56.iv.1.12	$56$	$2$	$2$	$193$	$22$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.jb.2.9	$56$	$2$	$2$	$193$	$22$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.kb.2.17	$56$	$2$	$2$	$193$	$28$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.kd.1.18	$56$	$2$	$2$	$193$	$28$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.bhl.1.12	$56$	$2$	$2$	$193$	$25$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.bhm.2.11	$56$	$2$	$2$	$193$	$25$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.bht.1.17	$56$	$2$	$2$	$193$	$29$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.bhu.1.18	$56$	$2$	$2$	$193$	$29$	$1^{16}\cdot2^{22}\cdot4^{5}\cdot6^{2}\cdot12$
56.5376.193-56.bpb.2.28	$56$	$2$	$2$	$193$	$3$	$2^{2}\cdot4^{4}\cdot8^{5}\cdot12\cdot16^{2}$
56.5376.193-56.bpf.2.28	$56$	$2$	$2$	$193$	$3$	$2^{2}\cdot4^{4}\cdot8^{5}\cdot12\cdot16^{2}$
56.5376.193-56.bpj.2.30	$56$	$2$	$2$	$193$	$3$	$2^{2}\cdot4^{4}\cdot8^{5}\cdot12\cdot16^{2}$
56.5376.193-56.bpn.2.30	$56$	$2$	$2$	$193$	$3$	$2^{2}\cdot4^{4}\cdot8^{5}\cdot12\cdot16^{2}$
56.8064.265-56.ca.1.47	$56$	$3$	$3$	$265$	$17$	$1^{52}\cdot2^{20}\cdot4^{6}\cdot6^{6}\cdot12^{2}$