LMFDB - Modular curve 56.192.11.eo.1

Invariants

Level:	$56$	$\SL_2$-level:	$56$	Newform level:	$784$
Index:	$192$	$\PSL_2$-index:	$192$
Genus:	$11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (none of which are rational)	Cusp widths	$2^{4}\cdot8^{2}\cdot14^{4}\cdot56^{2}$	Cusp orbits	$2^{6}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$3$
$\Q$-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:	56M11
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	56.192.11.139

Level structure

$\GL_2(\Z/56\Z)$-generators:	$\begin{bmatrix}3&53\\4&25\end{bmatrix}$, $\begin{bmatrix}11&52\\16&27\end{bmatrix}$, $\begin{bmatrix}25&49\\12&31\end{bmatrix}$, $\begin{bmatrix}35&22\\20&29\end{bmatrix}$, $\begin{bmatrix}43&35\\52&13\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	56.384.11-56.eo.1.1, 56.384.11-56.eo.1.2, 56.384.11-56.eo.1.3, 56.384.11-56.eo.1.4, 56.384.11-56.eo.1.5, 56.384.11-56.eo.1.6, 56.384.11-56.eo.1.7, 56.384.11-56.eo.1.8, 56.384.11-56.eo.1.9, 56.384.11-56.eo.1.10, 56.384.11-56.eo.1.11, 56.384.11-56.eo.1.12, 56.384.11-56.eo.1.13, 56.384.11-56.eo.1.14, 56.384.11-56.eo.1.15, 56.384.11-56.eo.1.16, 112.384.11-56.eo.1.1, 112.384.11-56.eo.1.2, 112.384.11-56.eo.1.3, 112.384.11-56.eo.1.4, 112.384.11-56.eo.1.5, 112.384.11-56.eo.1.6, 112.384.11-56.eo.1.7, 112.384.11-56.eo.1.8, 168.384.11-56.eo.1.1, 168.384.11-56.eo.1.2, 168.384.11-56.eo.1.3, 168.384.11-56.eo.1.4, 168.384.11-56.eo.1.5, 168.384.11-56.eo.1.6, 168.384.11-56.eo.1.7, 168.384.11-56.eo.1.8, 168.384.11-56.eo.1.9, 168.384.11-56.eo.1.10, 168.384.11-56.eo.1.11, 168.384.11-56.eo.1.12, 168.384.11-56.eo.1.13, 168.384.11-56.eo.1.14, 168.384.11-56.eo.1.15, 168.384.11-56.eo.1.16, 280.384.11-56.eo.1.1, 280.384.11-56.eo.1.2, 280.384.11-56.eo.1.3, 280.384.11-56.eo.1.4, 280.384.11-56.eo.1.5, 280.384.11-56.eo.1.6, 280.384.11-56.eo.1.7, 280.384.11-56.eo.1.8, 280.384.11-56.eo.1.9, 280.384.11-56.eo.1.10, 280.384.11-56.eo.1.11, 280.384.11-56.eo.1.12, 280.384.11-56.eo.1.13, 280.384.11-56.eo.1.14, 280.384.11-56.eo.1.15, 280.384.11-56.eo.1.16
Cyclic 56-isogeny field degree:	$2$
Cyclic 56-torsion field degree:	$48$
Full 56-torsion field degree:	$16128$

Jacobian

Conductor:	$2^{28}\cdot7^{17}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{11}$
Newforms:	14.2.a.a$^{2}$, 98.2.a.a$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 392.2.a.b$^{2}$, 392.2.a.d$^{2}$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

$ 0 $	$=$	$ x t - x v - s a $
	$=$	$x z - x w - z a$
	$=$	$t u - u v - r s - s a + s b$
	$=$	$x r + x a - x b - u a$
	$=$	$\cdots$

[x*t - x*v - s*a, x*z - x*w - z*a, t*u - u*v - r*s - s*a + s*b, x*r + x*a - x*b - u*a, x*u - x*r + x*b + z^2 + w*v, x*y - x*r - y*u - z*t, y*r - y*b + z*t - w*t + r*a, z*t - z*v - z*s + w*s, x*y + y*u - 2*z^2 + z*t - t*v + u*r + s^2, x^2 - x*u + y*u - y*r - z^2 + z*t + z*v + z*s - w*t + r*a, x^2 + x*y - y^2 + y*a - 2*z^2 + z*w, x^2 - x*u + z^2 - z*w + w^2 - w*v + u*a, x^2 + x*y - x*u + x*b + y*u - z^2 + z*t + w*v + u^2 + v*s - 2*r^2 + r*b, x^2 + x*y - x*a + x*b + y*u - z^2 - z*w + z*t + w^2 + u^2 - u*r + v^2 + v*s, x^2 + x*u - x*r + 2*y*a + u*r - v*s + r^2 - r*a + s^2, x*v + y*t - y*s + z*b - t*a, x*r - y*u + z^2 - z*t - 2*z*s + w*t, z*u - z*r - z*a + z*b - w*u, x*v - x*s + y*z + y*v - y*s + z*a + w*u - w*r, x*w - x*s + y*v + z*b - w*u - w*r, x*z - x*t - x*s + y*s - z*u - z*r, x^2 - x*y - y^2 + y*a - z*w + z*v + u*r - v*s, x*t + x*s - y*t - w*u + w*r - w*b + t*a + v*a, x*t + x*s - y*z + y*w - y*v - z*u + z*r - w*r + t*a, x^2 - x*u - x*r + x*b + z*w - z*t - z*v + z*s + w*v - w*s, x*y - x*u - x*a - y^2 + y*a + z^2 - z*v + z*s - u*a + v^2 - s^2, x*z + x*w + x*t + x*s + y*z - y*w + z*b - w*u - w*r + t*a - u*s - v*r, x*t + x*v - y*w - z*r + z*b - w*r - t*u + t*a - v*r - r*s, x*z + x*w + x*t + y*z - y*w - y*v + z*b - w*u - w*r - t*u + t*r + t*a - t*b + u*s + v*r, x*w + x*s + z*u - z*a + t*u - 2*t*r + t*b - u*v - u*s, z^2 - 2*z*w - z*s + w*v + u^2 - u*r + u*b - v^2 + s^2, x^2 - x*a - z^2 - z*v + z*s - w^2 - w*s - t*s - u^2 + v*s, x*y + x*a + x*b - z^2 - z*v + w^2 + w*v - w*s - t^2 + t*v - u^2 + u*r, y*z + y*s + z*r + z*a - w*r - 2*w*a + w*b - t*u - u*v + v*r - v*b + s*a - s*b, x*y + x*a + x*b + 2*z^2 + z*t + z*v - w*v - t^2 + t*v + u*b + 2*v^2 + r^2 + r*a - 2*r*b - s^2 + 2*a^2 - a*b + b^2, 2*x*z + x*w + x*s - 2*y*z - y*s + z*u + z*r + 2*z*a - w*u + t*u + s*b]

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=47$, and therefore no rational points.

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 28.96.5.k.1 :

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle -w$
$\displaystyle Z$	$=$	$\displaystyle v$
$\displaystyle W$	$=$	$\displaystyle t$
$\displaystyle T$	$=$	$\displaystyle s$

Equation of the image curve:

$0$	$=$	$ XZ-XW+XT+YT $
	$=$	$ 4XY+2Y^{2}+XZ+YZ+Z^{2}+XW+YW+2ZW-W^{2}+XT-T^{2} $
	$=$	$ 7X^{2}-XW-YW-2ZW+W^{2} $

Modular covers

Cover information

Click on a modular curve in the diagram to see information about it.

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
28.96.5.k.1	$28$	$2$	$2$	$5$	$1$	$1^{6}$
56.24.0.bm.1	$56$	$8$	$8$	$0$	$0$	full Jacobian
56.96.5.bn.1	$56$	$2$	$2$	$5$	$1$	$1^{6}$
56.96.5.bq.1	$56$	$2$	$2$	$5$	$1$	$1^{6}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
56.384.21.eh.1	$56$	$2$	$2$	$21$	$3$	$2^{5}$
56.384.21.eh.2	$56$	$2$	$2$	$21$	$3$	$2^{5}$
56.384.21.eh.3	$56$	$2$	$2$	$21$	$3$	$2^{5}$
56.384.21.eh.4	$56$	$2$	$2$	$21$	$3$	$2^{5}$
56.384.21.ei.1	$56$	$2$	$2$	$21$	$5$	$2^{5}$
56.384.21.ei.2	$56$	$2$	$2$	$21$	$5$	$2^{5}$
56.384.21.ei.3	$56$	$2$	$2$	$21$	$5$	$2^{5}$
56.384.21.ei.4	$56$	$2$	$2$	$21$	$5$	$2^{5}$
56.576.31.kc.1	$56$	$3$	$3$	$31$	$3$	$2^{10}$
56.576.31.kc.2	$56$	$3$	$3$	$31$	$3$	$2^{10}$
56.576.31.lu.1	$56$	$3$	$3$	$31$	$15$	$1^{20}$
56.1344.89.xg.1	$56$	$7$	$7$	$89$	$20$	$1^{48}\cdot2^{15}$
168.384.21.bjp.1	$168$	$2$	$2$	$21$	$?$	not computed
168.384.21.bjp.2	$168$	$2$	$2$	$21$	$?$	not computed
168.384.21.bjp.3	$168$	$2$	$2$	$21$	$?$	not computed
168.384.21.bjp.4	$168$	$2$	$2$	$21$	$?$	not computed
168.384.21.bjq.1	$168$	$2$	$2$	$21$	$?$	not computed
168.384.21.bjq.2	$168$	$2$	$2$	$21$	$?$	not computed
168.384.21.bjq.3	$168$	$2$	$2$	$21$	$?$	not computed
168.384.21.bjq.4	$168$	$2$	$2$	$21$	$?$	not computed
280.384.21.vf.1	$280$	$2$	$2$	$21$	$?$	not computed
280.384.21.vf.2	$280$	$2$	$2$	$21$	$?$	not computed
280.384.21.vf.3	$280$	$2$	$2$	$21$	$?$	not computed
280.384.21.vf.4	$280$	$2$	$2$	$21$	$?$	not computed
280.384.21.vg.1	$280$	$2$	$2$	$21$	$?$	not computed
280.384.21.vg.2	$280$	$2$	$2$	$21$	$?$	not computed
280.384.21.vg.3	$280$	$2$	$2$	$21$	$?$	not computed
280.384.21.vg.4	$280$	$2$	$2$	$21$	$?$	not computed

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