LMFDB - Modular curve 112.384.11-56.fa.3.28

Invariants

Level:	$112$	$\SL_2$-level:	$112$	Newform level:	$56$
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 $6$ are rational)	Cusp widths	$1^{2}\cdot2\cdot4\cdot7^{2}\cdot8^{2}\cdot14\cdot28\cdot56^{2}$	Cusp orbits	$1^{6}\cdot2^{3}$
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:	$6$
Rational CM points:	none

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}1&100\\10&35\end{bmatrix}$, $\begin{bmatrix}43&40\\98&97\end{bmatrix}$, $\begin{bmatrix}44&17\\95&78\end{bmatrix}$, $\begin{bmatrix}44&93\\15&10\end{bmatrix}$, $\begin{bmatrix}54&45\\1&98\end{bmatrix}$, $\begin{bmatrix}64&49\\49&8\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fa.3 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 w - x u - x v + y z $
	$=$	$x y + x t - x u - x v - y^{2} + y v + z w - z v + w^{2} + w t + t^{2} + u v$
	$=$	$x y - x z - x w + x t + x v - x s + y t - y u - y v - z w - z v + w t - t u + v r - v a - v b + r a - s a$
	$=$	$x^{2} + x y - x z + x w + x t + x u - x v - x r + y t - y u + y v + y r + 2 w t - t u - t v$
	$=$	$\cdots$

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

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:-2/3:-2/3:2/3:-4/3:0:-1:1)$, $(0:0:0:0:-1:1:-1:1:1:0:0)$, $(0:0:0:0:0:0:0:0:0:-1:1)$, $(0:0:0:0:0:1:0:0:0:0:0)$, $(0:0:0:0:0:0:1:0:0:0:0)$, $(0:0:0:0:-1/2:1/2:-1/2:3/2:1/2:0:1)$

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 $X_0(56)$ :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle -y$
$\displaystyle Z$	$=$	$\displaystyle -z$
$\displaystyle W$	$=$	$\displaystyle -w$
$\displaystyle T$	$=$	$\displaystyle u+v$

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 $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.192.5-56.bl.1.12	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

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Elliptic curves over $\Q$
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