LMFDB - Modular curve 112.384.11-56.ff.4.31

Invariants

Level:	$112$	$\SL_2$-level:	$112$	Newform level:	$224$
Index:	$384$	$\PSL_2$-index:	$192$
Genus:	$11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $4$ are rational)	Cusp widths	$1^{2}\cdot2\cdot4\cdot7^{2}\cdot8^{2}\cdot14\cdot28\cdot56^{2}$	Cusp orbits	$1^{4}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 11$
Rational cusps:	$4$
Rational CM points:	none

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}35&88\\38&29\end{bmatrix}$, $\begin{bmatrix}47&102\\24&13\end{bmatrix}$, $\begin{bmatrix}48&63\\67&100\end{bmatrix}$, $\begin{bmatrix}76&9\\57&28\end{bmatrix}$, $\begin{bmatrix}94&41\\61&18\end{bmatrix}$, $\begin{bmatrix}107&0\\90&73\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.ff.4 for the level structure with $-I$)
Cyclic 112-isogeny field degree:	$2$
Cyclic 112-torsion field degree:	$48$
Full 112-torsion field degree:	$129024$

Models

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

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

[x*v - x*a - y*v - w*a, x*s + y*r + w*u - w*v, x*r + x*a + z*u + z*v, x*u - x*r + x*b - y*u - y*v + w*u, x*u + x*s - y*v + y*a + z*u, x*v - x*s + y*v - z*a - t*u, x*u + x*r + w*u - w*r - w*s + w*b, x*u + x*v - x*r - y*v - z*v - z*s + w*v - t*v, x*v + x*a + y*u + z*u - z*r + w*u - w*s + w*a, x*s + x*a - y*u + y*s + z*v - w*u + w*a, x*u + y*r + y*s - y*b - z*r + w*u - w*s + w*a, x*r - y*r - y*s - y*a + z*r - z*b + w*u, y*r + y*a + w*u + w*v - t*u - t*v, x*w - x*t - y*z, x*u - x*s + y*u + y*s - z*u + z*s + w*r - w*s + w*b - t*r, x*r - x*a + y*s - z*v + z*r - w*s - w*a - t*r + t*b, y*v - y*r + y*s - z*u + z*r - z*s - w*u - w*s - w*a - t*u + t*s + t*a, u^2 + u*v - u*r - 2*u*s - v^2 + v*r + v*s + v*a - v*b + r*s + s^2 + s*a - s*b, u^2 + u*v - 2*u*r + u*b + v*r - v*b + r^2 - 2*r*a - r*b - s^2 - s*a + s*b + a*b, u^2 + 2*u*v - u*r + u*s - u*a - v*s + v*a + v*b - r*a - s^2 + a^2 + a*b, x^2 - x*z + y^2 + y*w - 2*w^2 + w*t, 2*x^2 + x*z - y^2 + y*w - y*t + z^2, 2*u^2 + u*v + 2*u*r + u*s - u*b - v^2 + v*r - v*a - v*b - r*s - 2*r*a + r*b - s^2 - s*a + 2*s*b + a*b - b^2, 3*u^2 + 2*u*v - 2*u*s + u*a - v*s - v*b + r^2 + r*a + 2*s*b - a*b - b^2, 2*x*y - x*w - 3*x*t + y*z - 2*z*w - 6*z*t - u^2 + u*v + u*r + u*s - v*r - v*s + s*b, 3*x*z + 3*y^2 - y*w + y*t + z^2 - 2*w^2 + 4*t^2 - u^2 + u*v + u*r + 2*u*s - 2*u*a - u*b + v*r - v*b - r*s + r*b - s^2 + s*a + a*b, 2*x^2 + 3*x*z + 3*y^2 - y*w + 3*y*t - z^2 - 2*w^2 - 2*w*t - 2*t^2 - u*v + 2*u*a + u*b - v*r + v*b - r*b - s*a + s*b - a*b, 3*x*z + y^2 - y*w + 5*y*t + z^2 + 4*w*t - 2*t^2 - u^2 + u*v + u*r - u*s + u*a - 2*v^2 + v*r + v*s - v*b - r*a + s^2, 4*x^2 + 4*y*t - 4*z^2 - 4*w*t + 2*t^2 - u*a + r*a + s*a, 2*x^2 + 2*x*z + 2*y^2 - 2*y*w - 4*y*t - 4*z^2 + u^2 - u*v - u*r - u*a + v*r + v*s - v*b + r*s - r*a - s*b + a*b, 4*x*y + 2*x*w + 4*x*t - 2*y*z - 4*z*w + 2*z*t - u*v + u*a - v*r + v*b + r*a - a*b, 4*x*y - 3*x*w - 5*x*t + 3*y*z - 4*z*w + 2*z*t - u*v + u*s - u*a + v^2 + v*r - v*a - v*b - r*s - r*a - s^2 - s*a + s*b + a*b, 2*x^2 - 2*x*z - 4*y^2 - 6*y*w - 2*z^2 - 4*w^2 - 2*w*t + 2*t^2 + u^2 - u*r + u*s - 2*u*a + v^2 - s^2 + a*b, 4*x^2 - 6*x*y - 4*x*z - x*w - x*t + 2*y^2 + y*z - 2*y*w + 2*y*t + 2*z^2 - 8*z*w + 4*z*t - 4*w*t + 2*t^2 - 2*u*r + u*b + v^2 - r*s - s^2 + s*b, 3*x^2 - 4*x*z + 9*x*w + 5*x*t + 2*y^2 + 5*y*z - 2*y*w + y*t + 3*z^2 - 3*w*t - 2*t^2 - 2*u*v - u*r + 2*u*a + u*b - 2*v*r + v*a + 3*v*b - r^2 - s*a - a*b, 5*x^2 + 10*x*y - 2*x*z - 7*x*w + x*t + 6*y^2 - 7*y*z - 8*y*w + y*t + 5*z^2 + 4*z*w - 2*z*t + 2*w^2 - w*t - a*b]

Singular plane model Singular plane model

$ 0 $

$=$

$ 196 x^{8} y^{2} z^{4} - 392 x^{8} y z^{5} + 196 x^{8} z^{6} + 112 x^{6} y^{6} z^{2} + 112 x^{6} y^{5} z^{3} + \cdots - 4 y z^{13} $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:0:1/2:0:-1/2:1:0:0)$, $(0:0:0:0:0:1/2:0:1/2:0:0:1)$, $(0:0:0:0:0:1/6:2/3:5/6:2/3:0:1)$, $(0:0:0:0:0:-1/2:1:-3/2:1:1:0)$

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 $X_0(56)$ :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle -y$
$\displaystyle Z$	$=$	$\displaystyle z$
$\displaystyle W$	$=$	$\displaystyle w$
$\displaystyle T$	$=$	$\displaystyle -t$

Equation of the image curve:

$0$	$=$	$ YZ+XW+XT $
	$=$	$ X^{2}+Y^{2}-XZ-YW-2W^{2}-WT $
	$=$	$ 2X^{2}-Y^{2}+XZ+Z^{2}-YW-YT $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.192.11.ff.4 :

$\displaystyle X$	$=$	$\displaystyle t$
$\displaystyle Y$	$=$	$\displaystyle v$
$\displaystyle Z$	$=$	$\displaystyle a$

Equation of the image curve:

$0$

$=$

$ 16X^{4}Y^{10}+128X^{4}Y^{9}Z-144X^{2}Y^{11}Z+16Y^{13}Z+112X^{6}Y^{6}Z^{2}-488X^{4}Y^{8}Z^{2}+512X^{2}Y^{10}Z^{2}-32Y^{12}Z^{2}+112X^{6}Y^{5}Z^{3}+168X^{4}Y^{7}Z^{3}-808X^{2}Y^{9}Z^{3}-8Y^{11}Z^{3}+196X^{8}Y^{2}Z^{4}-756X^{6}Y^{4}Z^{4}+865X^{4}Y^{6}Z^{4}+888X^{2}Y^{8}Z^{4}+72Y^{10}Z^{4}-392X^{8}YZ^{5}+364X^{6}Y^{3}Z^{5}-1078X^{4}Y^{5}Z^{5}-749X^{2}Y^{7}Z^{5}-47Y^{9}Z^{5}+196X^{8}Z^{6}+448X^{6}Y^{2}Z^{6}+751X^{4}Y^{4}Z^{6}+542X^{2}Y^{6}Z^{6}-24Y^{8}Z^{6}-84X^{6}YZ^{7}-344X^{4}Y^{3}Z^{7}-227X^{2}Y^{5}Z^{7}+41Y^{7}Z^{7}-196X^{6}Z^{8}-233X^{4}Y^{2}Z^{8}-204X^{2}Y^{4}Z^{8}-28Y^{6}Z^{8}+126X^{4}YZ^{9}+177X^{2}Y^{3}Z^{9}+15Y^{5}Z^{9}+105X^{4}Z^{10}+82X^{2}Y^{2}Z^{10}+8Y^{4}Z^{10}-41X^{2}YZ^{11}-13Y^{3}Z^{11}-28X^{2}Z^{12}+4Y^{2}Z^{12}-4YZ^{13} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.192.5-56.bl.1.9	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

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