Modular curve 56.1344.93.l.1

• Label: 56.1344.93.l.1
• Level: $56$
• Index: $1344$
• Genus: $93$
• Analytic rank: $7$
• Cusps: $40$
• $\Q$-cusps: $4$

Invariants

Level:	$56$		$\SL_2$-level:	$56$		Newform level:	$784$
Index:	$1344$		$\PSL_2$-index:	$1344$
Genus:	$93 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$
Cusps:	$40$ (of which $4$ are rational)		Cusp widths	$28^{32}\cdot56^{8}$		Cusp orbits	$1^{4}\cdot2^{3}\cdot3^{4}\cdot6^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$7$
$\Q$-gonality:	$14 \le \gamma \le 28$
$\overline{\Q}$-gonality:	$14 \le \gamma \le 28$
Rational cusps:	$4$
Rational CM points:	none

Other labels

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

56.1344.93.2

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}1&12\\12&41\end{bmatrix}$, $\begin{bmatrix}9&4\\8&47\end{bmatrix}$, $\begin{bmatrix}9&36\\44&45\end{bmatrix}$, $\begin{bmatrix}15&44\\4&17\end{bmatrix}$, $\begin{bmatrix}33&20\\40&25\end{bmatrix}$, $\begin{bmatrix}33&28\\12&23\end{bmatrix}$, $\begin{bmatrix}39&36\\44&45\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.2688.93-56.l.1.1, 56.2688.93-56.l.1.2, 56.2688.93-56.l.1.3, 56.2688.93-56.l.1.4, 56.2688.93-56.l.1.5, 56.2688.93-56.l.1.6, 56.2688.93-56.l.1.7, 56.2688.93-56.l.1.8, 56.2688.93-56.l.1.9, 56.2688.93-56.l.1.10, 56.2688.93-56.l.1.11, 56.2688.93-56.l.1.12, 56.2688.93-56.l.1.13, 56.2688.93-56.l.1.14, 56.2688.93-56.l.1.15, 56.2688.93-56.l.1.16, 56.2688.93-56.l.1.17, 56.2688.93-56.l.1.18, 56.2688.93-56.l.1.19, 56.2688.93-56.l.1.20, 56.2688.93-56.l.1.21, 56.2688.93-56.l.1.22, 56.2688.93-56.l.1.23, 56.2688.93-56.l.1.24, 56.2688.93-56.l.1.25, 56.2688.93-56.l.1.26, 56.2688.93-56.l.1.27, 56.2688.93-56.l.1.28, 56.2688.93-56.l.1.29, 56.2688.93-56.l.1.30, 56.2688.93-56.l.1.31, 56.2688.93-56.l.1.32, 56.2688.93-56.l.1.33, 56.2688.93-56.l.1.34, 56.2688.93-56.l.1.35, 56.2688.93-56.l.1.36, 56.2688.93-56.l.1.37, 56.2688.93-56.l.1.38, 56.2688.93-56.l.1.39, 56.2688.93-56.l.1.40
Cyclic 56-isogeny field degree:	$4$
Cyclic 56-torsion field degree:	$48$
Full 56-torsion field degree:	$2304$

Jacobian

Conductor:	$2^{258}\cdot7^{163}$
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$^{2}$, 56.2.b.a$^{2}$, 56.2.b.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$^{3}$, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 392.2.b.e$^{2}$, 392.2.b.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

Rational points

This modular curve has 4 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
$X_{\mathrm{sp}}^+(7)$	$7$	$48$	$48$	$0$	$0$	full Jacobian
8.48.0.c.1	$8$	$28$	$28$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.48.0.c.1	$8$	$28$	$28$	$0$	$0$	full Jacobian
28.672.45.b.1	$28$	$2$	$2$	$45$	$7$	$2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$
56.672.45.bb.1	$56$	$2$	$2$	$45$	$1$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.672.45.bb.2	$56$	$2$	$2$	$45$	$1$	$1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.2688.185.m.1	$56$	$2$	$2$	$185$	$18$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.q.1	$56$	$2$	$2$	$185$	$34$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.bm.1	$56$	$2$	$2$	$185$	$16$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.bq.1	$56$	$2$	$2$	$185$	$30$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.cp.1	$56$	$2$	$2$	$185$	$23$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.ct.1	$56$	$2$	$2$	$185$	$24$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.dn.1	$56$	$2$	$2$	$185$	$19$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.dr.1	$56$	$2$	$2$	$185$	$24$	$1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.193.px.1	$56$	$2$	$2$	$193$	$29$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.193.px.2	$56$	$2$	$2$	$193$	$29$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.193.py.1	$56$	$2$	$2$	$193$	$31$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.193.py.2	$56$	$2$	$2$	$193$	$31$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.193.pz.1	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.pz.2	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.qa.1	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.qa.2	$56$	$2$	$2$	$193$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.qb.1	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.qb.2	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.qc.1	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.qc.2	$56$	$2$	$2$	$193$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$
56.2688.193.qd.1	$56$	$2$	$2$	$193$	$26$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.193.qd.2	$56$	$2$	$2$	$193$	$26$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.193.qe.1	$56$	$2$	$2$	$193$	$22$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.193.qe.2	$56$	$2$	$2$	$193$	$22$	$1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.2688.201.ey.1	$56$	$2$	$2$	$201$	$29$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.2688.201.ez.1	$56$	$2$	$2$	$201$	$22$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.2688.201.fa.1	$56$	$2$	$2$	$201$	$31$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.2688.201.fb.1	$56$	$2$	$2$	$201$	$26$	$1^{16}\cdot2^{20}\cdot4^{6}\cdot12\cdot16$
56.2688.201.fc.1	$56$	$2$	$2$	$201$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.2688.201.fd.1	$56$	$2$	$2$	$201$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.2688.201.fe.1	$56$	$2$	$2$	$201$	$24$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.2688.201.ff.1	$56$	$2$	$2$	$201$	$32$	$1^{32}\cdot2^{16}\cdot4^{6}\cdot8\cdot12$
56.4032.277.bm.1	$56$	$3$	$3$	$277$	$28$	$1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$