Modular curve 56.252.16.b.1

• Label: 56.252.16.b.1
• Level: $56$
• Index: $252$
• Genus: $16$
• Analytic rank: $0$
• Cusps: $12$
• $\Q$-cusps: $0$

Invariants

Level:	$56$		$\SL_2$-level:	$28$		Newform level:	$3136$
Index:	$252$		$\PSL_2$-index:	$252$
Genus:	$16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (none of which are rational)		Cusp widths	$14^{6}\cdot28^{6}$		Cusp orbits	$3^{2}\cdot6$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$5 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 8$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	28C16
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	56.252.16.2

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}5&52\\4&5\end{bmatrix}$, $\begin{bmatrix}13&16\\12&27\end{bmatrix}$, $\begin{bmatrix}33&14\\28&51\end{bmatrix}$, $\begin{bmatrix}49&50\\50&35\end{bmatrix}$, $\begin{bmatrix}51&42\\14&47\end{bmatrix}$, $\begin{bmatrix}55&20\\48&15\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.504.16-56.b.1.1, 56.504.16-56.b.1.2, 56.504.16-56.b.1.3, 56.504.16-56.b.1.4, 56.504.16-56.b.1.5, 56.504.16-56.b.1.6, 56.504.16-56.b.1.7, 56.504.16-56.b.1.8, 56.504.16-56.b.1.9, 56.504.16-56.b.1.10, 56.504.16-56.b.1.11, 56.504.16-56.b.1.12, 56.504.16-56.b.1.13, 56.504.16-56.b.1.14, 56.504.16-56.b.1.15, 56.504.16-56.b.1.16, 168.504.16-56.b.1.1, 168.504.16-56.b.1.2, 168.504.16-56.b.1.3, 168.504.16-56.b.1.4, 168.504.16-56.b.1.5, 168.504.16-56.b.1.6, 168.504.16-56.b.1.7, 168.504.16-56.b.1.8, 168.504.16-56.b.1.9, 168.504.16-56.b.1.10, 168.504.16-56.b.1.11, 168.504.16-56.b.1.12, 168.504.16-56.b.1.13, 168.504.16-56.b.1.14, 168.504.16-56.b.1.15, 168.504.16-56.b.1.16, 280.504.16-56.b.1.1, 280.504.16-56.b.1.2, 280.504.16-56.b.1.3, 280.504.16-56.b.1.4, 280.504.16-56.b.1.5, 280.504.16-56.b.1.6, 280.504.16-56.b.1.7, 280.504.16-56.b.1.8, 280.504.16-56.b.1.9, 280.504.16-56.b.1.10, 280.504.16-56.b.1.11, 280.504.16-56.b.1.12, 280.504.16-56.b.1.13, 280.504.16-56.b.1.14, 280.504.16-56.b.1.15, 280.504.16-56.b.1.16
Cyclic 56-isogeny field degree:	$32$
Cyclic 56-torsion field degree:	$768$
Full 56-torsion field degree:	$12288$

Jacobian

Conductor:	$2^{64}\cdot7^{32}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{4}\cdot2^{6}$
Newforms:	98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 3136.2.a.bc, 3136.2.a.bm, 3136.2.a.bp, 3136.2.a.bs, 3136.2.a.j, 3136.2.a.s

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
$X_{\mathrm{ns}}^+(7)$	$7$	$12$	$12$	$0$	$0$	full Jacobian
8.12.0.b.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
8.12.0.b.1	$8$	$21$	$21$	$0$	$0$	full Jacobian
14.126.7.a.1	$14$	$2$	$2$	$7$	$0$	$1^{3}\cdot2^{3}$
56.126.6.c.1	$56$	$2$	$2$	$6$	$0$	$1^{2}\cdot2^{4}$
56.126.7.d.1	$56$	$2$	$2$	$7$	$0$	$1^{3}\cdot2^{3}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.504.31.d.1	$56$	$2$	$2$	$31$	$10$	$1^{13}\cdot2$
56.504.31.f.1	$56$	$2$	$2$	$31$	$9$	$1^{13}\cdot2$
56.504.31.j.1	$56$	$2$	$2$	$31$	$6$	$1^{13}\cdot2$
56.504.31.l.1	$56$	$2$	$2$	$31$	$3$	$1^{13}\cdot2$
56.504.31.bb.1	$56$	$2$	$2$	$31$	$4$	$1^{13}\cdot2$
56.504.31.bd.1	$56$	$2$	$2$	$31$	$4$	$1^{13}\cdot2$
56.504.31.bh.1	$56$	$2$	$2$	$31$	$7$	$1^{13}\cdot2$
56.504.31.bj.1	$56$	$2$	$2$	$31$	$5$	$1^{13}\cdot2$
56.504.34.b.1	$56$	$2$	$2$	$34$	$4$	$1^{6}\cdot2^{6}$
56.504.34.c.1	$56$	$2$	$2$	$34$	$10$	$1^{6}\cdot2^{6}$
56.504.34.i.1	$56$	$2$	$2$	$34$	$6$	$1^{6}\cdot2^{6}$
56.504.34.j.1	$56$	$2$	$2$	$34$	$8$	$1^{6}\cdot2^{6}$
56.504.34.bf.1	$56$	$2$	$2$	$34$	$5$	$1^{14}\cdot2^{2}$
56.504.34.bg.1	$56$	$2$	$2$	$34$	$8$	$1^{14}\cdot2^{2}$
56.504.34.bm.1	$56$	$2$	$2$	$34$	$3$	$1^{14}\cdot2^{2}$
56.504.34.bn.1	$56$	$2$	$2$	$34$	$12$	$1^{14}\cdot2^{2}$