Modular curve 56.2688.93-56.a.1.21

• Label: 56.2688.93-56.a.1.21
• Level: $56$
• Index: $2688$
• Genus: $93$
• Analytic rank: $7$
• Cusps: $40$
• $\Q$-cusps: $0$

Invariants

Level:	$56$		$\SL_2$-level:	$56$		Newform level:	$1568$
Index:	$2688$		$\PSL_2$-index:	$1344$
Genus:	$93 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$
Cusps:	$40$ (none of which are rational)		Cusp widths	$28^{32}\cdot56^{8}$		Cusp orbits	$2^{5}\cdot6^{5}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$7$
$\Q$-gonality:	$14 \le \gamma \le 48$
$\overline{\Q}$-gonality:	$14 \le \gamma \le 28$
Rational cusps:	$0$
Rational CM points:	none

Other labels

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

56.2688.93.72

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}1&0\\20&17\end{bmatrix}$, $\begin{bmatrix}13&0\\28&41\end{bmatrix}$, $\begin{bmatrix}29&40\\44&41\end{bmatrix}$, $\begin{bmatrix}31&28\\24&11\end{bmatrix}$, $\begin{bmatrix}33&28\\42&47\end{bmatrix}$, $\begin{bmatrix}41&0\\4&33\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.1344.93.a.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree:	$4$
Cyclic 56-torsion field degree:	$48$
Full 56-torsion field degree:	$1152$

Jacobian

Conductor:	$2^{354}\cdot7^{163}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{21}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$
Newforms:	14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 98.2.a.b$^{4}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 224.2.b.a$^{2}$, 224.2.b.b$^{2}$, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.b.f$^{2}$, 1568.2.b.g$^{2}$

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=31,79,223$, 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
$X_{\mathrm{sp}}^+(7)$	$7$	$96$	$48$	$0$	$0$	full Jacobian
8.96.0-8.a.1.4	$8$	$28$	$28$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.96.0-8.a.1.4	$8$	$28$	$28$	$0$	$0$	full Jacobian
56.1344.45-28.b.1.1	$56$	$2$	$2$	$45$	$7$	$2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$
56.1344.45-28.b.1.40	$56$	$2$	$2$	$45$	$7$	$2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$
56.1344.45-56.u.1.18	$56$	$2$	$2$	$45$	$1$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.1344.45-56.u.1.19	$56$	$2$	$2$	$45$	$1$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.1344.45-56.u.1.46	$56$	$2$	$2$	$45$	$1$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.1344.45-56.u.1.47	$56$	$2$	$2$	$45$	$1$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.5376.185-56.e.1.22	$56$	$2$	$2$	$185$	$18$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.185-56.f.1.14	$56$	$2$	$2$	$185$	$34$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.185-56.be.1.18	$56$	$2$	$2$	$185$	$16$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.185-56.bf.1.12	$56$	$2$	$2$	$185$	$30$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.185-56.ci.1.11	$56$	$2$	$2$	$185$	$23$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.185-56.cj.1.9	$56$	$2$	$2$	$185$	$24$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.185-56.dg.1.11	$56$	$2$	$2$	$185$	$19$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.185-56.dh.1.9	$56$	$2$	$2$	$185$	$24$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lc.1.21	$56$	$2$	$2$	$193$	$29$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.lc.2.1	$56$	$2$	$2$	$193$	$29$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.lc.2.8	$56$	$2$	$2$	$193$	$29$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.ld.1.24	$56$	$2$	$2$	$193$	$31$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.ld.2.3	$56$	$2$	$2$	$193$	$31$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.ld.2.6	$56$	$2$	$2$	$193$	$31$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.le.1.17	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.le.2.9	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.le.2.16	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lf.1.17	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lf.2.10	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lf.2.15	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lg.1.17	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lg.2.12	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lg.2.13	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lh.1.17	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lh.2.11	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.lh.2.14	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.5376.193-56.li.1.24	$56$	$2$	$2$	$193$	$26$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.li.2.2	$56$	$2$	$2$	$193$	$26$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.li.2.7	$56$	$2$	$2$	$193$	$26$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.lj.1.21	$56$	$2$	$2$	$193$	$22$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.lj.2.1	$56$	$2$	$2$	$193$	$22$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.193-56.lj.2.8	$56$	$2$	$2$	$193$	$22$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.5376.201-56.z.1.22	$56$	$2$	$2$	$201$	$29$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.5376.201-56.ba.1.24	$56$	$2$	$2$	$201$	$22$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.5376.201-56.bb.1.21	$56$	$2$	$2$	$201$	$31$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.5376.201-56.bc.1.23	$56$	$2$	$2$	$201$	$26$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.5376.201-56.bd.1.23	$56$	$2$	$2$	$201$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.5376.201-56.be.1.21	$56$	$2$	$2$	$201$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.5376.201-56.bf.1.24	$56$	$2$	$2$	$201$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.5376.201-56.bg.1.22	$56$	$2$	$2$	$201$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.8064.277-56.z.1.13	$56$	$3$	$3$	$277$	$28$	$1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$