LMFDB - Modular curve 112.384.11-56.fh.1.2

Invariants

Level:	$112$	$\SL_2$-level:	$112$	Newform level:	$112$
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	$2^{4}\cdot8^{2}\cdot14^{4}\cdot56^{2}$	Cusp orbits	$1^{6}\cdot2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 11$
Rational cusps:	$6$
Rational CM points:	none

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}5&102\\28&71\end{bmatrix}$, $\begin{bmatrix}34&95\\53&76\end{bmatrix}$, $\begin{bmatrix}50&55\\23&18\end{bmatrix}$, $\begin{bmatrix}55&108\\88&35\end{bmatrix}$, $\begin{bmatrix}67&92\\62&9\end{bmatrix}$, $\begin{bmatrix}84&37\\75&94\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fh.1 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 $	$=$	$ y u - z w - t u $
	$=$	$x u - x v - x b - y u - y b - u s + s b$
	$=$	$2 x^{2} + x y + x w - x t + x a + r s + r a$
	$=$	$x v + 2 y z + y u + y v + y b - u a + v a + a b$
	$=$	$\cdots$

[y*u - z*w - t*u, x*u - x*v - x*b - y*u - y*b - u*s + s*b, 2*x^2 + x*y + x*w - x*t + x*a + r*s + r*a, x*v + 2*y*z + y*u + y*v + y*b - u*a + v*a + a*b, 2*x*z - x*b - z*w - w*u - u*a + v*s + v*a - r*b + s*b + a*b, x^2 - x*s - y^2 + 2*y*t + v^2 - s*a, x^2 - x*s + y*a + u*v + v^2 - v*b - s*a, x*v + y*u + y*b - w*u + t*b - r*b + a*b, x*v + z*w - z*a + t*b, x*z - z*w - z*s - t*b, x*z - x*v - y*z - y*u - y*b + z*a - t*v - a*b, x^2 + x*y - x*t - x*s - t*s - u*v + v^2, x*t - y*s + t*s + v^2 + v*b - s*a, x^2 + x*w + x*s - t*s - t*a + r*s + r*a, y*z + z*w + z*a + t*v + t*b, x*s + x*a - t*a - u^2 + u*b + v*b + r*a, x*z + x*v + y*u + z*s - t*v + v*r, x*u - y*u - w*v - u*s - v*s + a*b, x^2 + x*y - x*s + z*u - z*b + t*a + v^2, x*v + z*t - z*r + t*u, x^2 + x*y - x*r + x*a + t*a - u^2 + u*v + u*b - v*b + r*s, x*y - x*t - x*r + x*a + z*v - v^2 - v*b + r*s, x^2 - x*s - z*u + z*v + z*b + t*a + v^2, x*y + x*w - x*t + z*v + t*s - u^2 + u*v + r*s + b^2, x^2 + x*w - x*t - x*s + y*w + y*t - y*r + y*s + y*a + z*v - w*s - t*a - u*v + v^2, x*y - x*s + y*a + w*s + w*a - t*s - u*b + r*s - s*a + b^2, x^2 - x*s + x*a + y*t - y*r + y*s + y*a + w*t - t^2 + t*r, z*a + w*v + w*b - t*b + u*r - u*s - u*a + v*a + s*b, z*u + z*b + w*t + t^2 - t*r - t*a - u*v + v*b, x*w - x*t + x*a + y*w + y*s - y*a + w^2 - w*t + t*r - v^2 - r^2 + r*s + r*a, x*t - x*r - x*a - y*s + y*a - w^2 - w*t + w*r - w*s + u^2 + u*b + v^2, x^2 + x*y - x*t + y*t + y*r - y*s - y*a + z^2 - z*v - w*t - w*r + t*a + v^2 - r^2 + r*a, x^2 - x*r + y*t - z^2 - w*r + t^2 + t*r + u*v - v*b - r^2 + r*s + r*a, x*u - x*b + y*u + y*v + y*b + z*s + w*u - w*v - w*b + t*v + u*r + u*s - r*b - s*b + a*b, x^2 - x*t + x*r - y*t - y*r + 3*z^2 - z*u - z*b + 2*t^2 + 2*t*r + t*s - t*a - v^2 + r*s, x*u - x*v + x*b + y*z + y*u + y*v + y*b + z*w + 4*z*t + 3*z*r - z*a + w*v - t*u - t*b - u*r + v*r + r*b]

Singular plane model Singular plane model

$ 0 $

$=$

$ 8 x^{6} y^{6} z + 8 x^{5} y^{8} - 56 x^{5} y^{4} z^{4} - 34 x^{4} y^{8} z - 22 x^{4} y^{6} z^{3} + \cdots + 14406 z^{13} $

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
$(0:0:0:-1:0:1:0:1:0:0:1)$, $(0:0:0:0:0:0:0:0:0:1:0)$, $(0:0:0:1:0:1:0:-1:0:0:1)$, $(0:0:0:0:0:0:0:0:1:0:0)$, $(2/3:0:0:-1:0:-1/3:0:-1/3:2/3:0:1)$, $(-2/3:0:0:1:0:-1/3:0:1/3:-2/3:0:1)$

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 28.96.4.c.2 :

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle -z-v$
$\displaystyle Z$	$=$	$\displaystyle -u$
$\displaystyle W$	$=$	$\displaystyle z+v+b$

Equation of the image curve:

$0$	$=$	$ X^{2}+4XY+2Y^{2}+3XZ-YZ-3Z^{2}+XW+YW-2ZW+W^{2} $
	$=$	$ 5X^{3}+X^{2}Y-2Y^{3}+X^{2}Z-3XYZ-XZ^{2}+2YZ^{2}+X^{2}W-Y^{2}W-2XZW-YW^{2} $

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

$\displaystyle X$	$=$	$\displaystyle b$
$\displaystyle Y$	$=$	$\displaystyle x$
$\displaystyle Z$	$=$	$\displaystyle z$

Equation of the image curve:

$0$

$=$

$ 8X^{5}Y^{8}-18X^{3}Y^{10}+8X^{6}Y^{6}Z-34X^{4}Y^{8}Z+54X^{2}Y^{10}Z+100X^{3}Y^{8}Z^{2}-153XY^{10}Z^{2}-22X^{4}Y^{6}Z^{3}-105X^{2}Y^{8}Z^{3}+234Y^{10}Z^{3}-56X^{5}Y^{4}Z^{4}+154X^{3}Y^{6}Z^{4}-321XY^{8}Z^{4}-98X^{4}Y^{4}Z^{5}-215X^{2}Y^{6}Z^{5}+703Y^{8}Z^{5}+1316X^{3}Y^{4}Z^{6}-2226XY^{6}Z^{6}+784X^{4}Y^{2}Z^{7}-3185X^{2}Y^{4}Z^{7}+4116Y^{6}Z^{7}+686X^{3}Y^{2}Z^{8}-889XY^{4}Z^{8}-5194X^{2}Y^{2}Z^{9}+7399Y^{4}Z^{9}-5145XY^{2}Z^{10}-9604X^{2}Z^{11}+16170Y^{2}Z^{11}+9604XZ^{12}+14406Z^{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.1.13	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

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