LMFDB - Modular curve 112.384.11-56.fd.2.25

Invariants

Level:	$112$	$\SL_2$-level:	$112$	Newform level:	$392$
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:	$2 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 11$
Rational cusps:	$4$
Rational CM points:	none

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}22&85\\103&60\end{bmatrix}$, $\begin{bmatrix}34&95\\3&70\end{bmatrix}$, $\begin{bmatrix}50&75\\111&70\end{bmatrix}$, $\begin{bmatrix}52&89\\11&18\end{bmatrix}$, $\begin{bmatrix}57&22\\14&9\end{bmatrix}$, $\begin{bmatrix}81&36\\108&9\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fd.2 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 t - y z $
	$=$	$x z + x b - y z + z v$
	$=$	$x u + x v - x b - z u$
	$=$	$y z - y w - y t + y b + w v + t v$
	$=$	$\cdots$

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

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
$(0:0:0:0:0:-1/2:0:1/2:1:1:0)$, $(0:0:0:0:0:0:1:0:1:0:1)$, $(0:0:0:0:0:1:0:1:0:0:0)$, $(0:0:0:0:0:0:1:0:-1:0:1)$

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 $X_0(56)$ :

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

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 $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.48.0-56.bv.1.1	$112$	$8$	$8$	$0$	$?$
112.192.5-56.bl.1.14	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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