Modular curve 56.1008.70.l.1

• Label: 56.1008.70.l.1
• Level: $56$
• Index: $1008$
• Genus: $70$
• Analytic rank: $6$
• Cusps: $30$
• $\Q$-cusps: $0$

Invariants

Level:	$56$		$\SL_2$-level:	$56$		Newform level:	$784$
Index:	$1008$		$\PSL_2$-index:	$1008$
Genus:	$70 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 30 }{2}$
Cusps:	$30$ (none of which are rational)		Cusp widths	$28^{24}\cdot56^{6}$		Cusp orbits	$3^{4}\cdot6^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$6$
$\Q$-gonality:	$10 \le \gamma \le 21$
$\overline{\Q}$-gonality:	$10 \le \gamma \le 21$
Rational cusps:	$0$
Rational CM points:	none

Other labels

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

56.1008.70.2

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}7&36\\48&35\end{bmatrix}$, $\begin{bmatrix}9&40\\8&33\end{bmatrix}$, $\begin{bmatrix}11&0\\12&31\end{bmatrix}$, $\begin{bmatrix}19&12\\38&37\end{bmatrix}$, $\begin{bmatrix}27&40\\48&15\end{bmatrix}$, $\begin{bmatrix}31&44\\16&51\end{bmatrix}$, $\begin{bmatrix}39&36\\18&45\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.2016.70-56.l.1.1, 56.2016.70-56.l.1.2, 56.2016.70-56.l.1.3, 56.2016.70-56.l.1.4, 56.2016.70-56.l.1.5, 56.2016.70-56.l.1.6, 56.2016.70-56.l.1.7, 56.2016.70-56.l.1.8, 56.2016.70-56.l.1.9, 56.2016.70-56.l.1.10, 56.2016.70-56.l.1.11, 56.2016.70-56.l.1.12, 56.2016.70-56.l.1.13, 56.2016.70-56.l.1.14, 56.2016.70-56.l.1.15, 56.2016.70-56.l.1.16, 56.2016.70-56.l.1.17, 56.2016.70-56.l.1.18, 56.2016.70-56.l.1.19, 56.2016.70-56.l.1.20, 56.2016.70-56.l.1.21, 56.2016.70-56.l.1.22, 56.2016.70-56.l.1.23, 56.2016.70-56.l.1.24, 56.2016.70-56.l.1.25, 56.2016.70-56.l.1.26, 56.2016.70-56.l.1.27, 56.2016.70-56.l.1.28, 56.2016.70-56.l.1.29, 56.2016.70-56.l.1.30, 56.2016.70-56.l.1.31, 56.2016.70-56.l.1.32, 56.2016.70-56.l.1.33, 56.2016.70-56.l.1.34, 56.2016.70-56.l.1.35, 56.2016.70-56.l.1.36, 56.2016.70-56.l.1.37, 56.2016.70-56.l.1.38, 56.2016.70-56.l.1.39, 56.2016.70-56.l.1.40
Cyclic 56-isogeny field degree:	$16$
Cyclic 56-torsion field degree:	$192$
Full 56-torsion field degree:	$3072$

Jacobian

Conductor:	$2^{194}\cdot7^{140}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{10}\cdot2^{12}\cdot6^{2}\cdot12^{2}$
Newforms:	98.2.a.b$^{4}$, 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 no $\Q_p$ points for $p=3,5,11,23,37,67,103,149$, 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
$X_{\mathrm{ns}}^+(7)$	$7$	$48$	$48$	$0$	$0$	full Jacobian
8.48.0.c.1	$8$	$21$	$21$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.48.0.c.1	$8$	$21$	$21$	$0$	$0$	full Jacobian
28.504.34.b.1	$28$	$2$	$2$	$34$	$6$	$6^{2}\cdot12^{2}$
56.504.34.z.1	$56$	$2$	$2$	$34$	$1$	$1^{6}\cdot2^{6}\cdot6\cdot12$
56.504.34.z.2	$56$	$2$	$2$	$34$	$1$	$1^{6}\cdot2^{6}\cdot6\cdot12$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.2016.139.h.1	$56$	$2$	$2$	$139$	$16$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.139.l.1	$56$	$2$	$2$	$139$	$30$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.139.bi.1	$56$	$2$	$2$	$139$	$12$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.139.bm.1	$56$	$2$	$2$	$139$	$21$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.139.cj.1	$56$	$2$	$2$	$139$	$19$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.139.cn.1	$56$	$2$	$2$	$139$	$18$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.139.dh.1	$56$	$2$	$2$	$139$	$15$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.139.dl.1	$56$	$2$	$2$	$139$	$21$	$1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.2016.145.pd.1	$56$	$2$	$2$	$145$	$23$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.145.pd.2	$56$	$2$	$2$	$145$	$23$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.145.pe.1	$56$	$2$	$2$	$145$	$24$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.145.pe.2	$56$	$2$	$2$	$145$	$24$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.145.pf.1	$56$	$2$	$2$	$145$	$21$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.pf.2	$56$	$2$	$2$	$145$	$21$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.pg.1	$56$	$2$	$2$	$145$	$18$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.pg.2	$56$	$2$	$2$	$145$	$18$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.ph.1	$56$	$2$	$2$	$145$	$25$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.ph.2	$56$	$2$	$2$	$145$	$25$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.pi.1	$56$	$2$	$2$	$145$	$26$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.pi.2	$56$	$2$	$2$	$145$	$26$	$1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.2016.145.pj.1	$56$	$2$	$2$	$145$	$20$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.145.pj.2	$56$	$2$	$2$	$145$	$20$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.145.pk.1	$56$	$2$	$2$	$145$	$19$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.145.pk.2	$56$	$2$	$2$	$145$	$19$	$1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.2016.151.ex.1	$56$	$2$	$2$	$151$	$23$	$1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.2016.151.ey.1	$56$	$2$	$2$	$151$	$19$	$1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.2016.151.ez.1	$56$	$2$	$2$	$151$	$24$	$1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.2016.151.fa.1	$56$	$2$	$2$	$151$	$20$	$1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.2016.151.fb.1	$56$	$2$	$2$	$151$	$18$	$1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$
56.2016.151.fc.1	$56$	$2$	$2$	$151$	$25$	$1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$
56.2016.151.fd.1	$56$	$2$	$2$	$151$	$21$	$1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$
56.2016.151.fe.1	$56$	$2$	$2$	$151$	$26$	$1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$