LMFDB - Modular curve 112.384.11-56.ff.1.30

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:	$\begin{bmatrix}20&107\\99&84\end{bmatrix}$, $\begin{bmatrix}27&56\\14&13\end{bmatrix}$, $\begin{bmatrix}54&59\\49&64\end{bmatrix}$, $\begin{bmatrix}54&75\\43&86\end{bmatrix}$, $\begin{bmatrix}98&57\\75&24\end{bmatrix}$, $\begin{bmatrix}99&0\\76&79\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.ff.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 $	$=$	$ t a - t b - r s $
	$=$	$x t - y t - u a - v a$
	$=$	$z t - z s + u r + u a + v r + v a$
	$=$	$x^{2} - y^{2} - z w - z v - t b + r^{2} - a^{2}$
	$=$	$\cdots$

[t*a - t*b - r*s, x*t - y*t - u*a - v*a, z*t - z*s + u*r + u*a + v*r + v*a, x^2 - y^2 - z*w - z*v - t*b + r^2 - a^2, x*y + x*u - y^2 - 2*y*z + y*v - z*u - r*a - s*a, x^2 + x*z - y^2 - y*z + 2*z*v - r^2 + a^2 - a*b, x^2 - x*y - x*z + x*u + x*v - y*z + z*w - z*u + z*v - r^2 - r*s, x^2 - x*z - y^2 - y*z + z^2 + z*v - t*r - t*s - t*a - a^2, x*r + x*s + y*s - y*a - 2*z*t + z*r - u*a, x*t + y*r - y*s - y*a + y*b + z*t + z*s - u*r + u*s - v*r + v*s - v*a, x^2 - x*y + x*z + y*z + y*u + y*v - z*v - u*v - v^2 + r^2 + r*s, x*t - x*r + x*a + y*t + z*r - t*v, x*t - y*t + z*t - z*s + t*v - u*r + u*s + u*a + v*s - v*a + v*b, x*t + x*s - y*t - y*s + t*u + t*v - u*r + v*s, x*t - x*r + y*r - y*s + z*t + z*s + w*a - u*r - u*a + v*s, x*t - y*s - z*t + z*s - w*s - t*u - u*r, x*t + x*b + y*t + z*t + z*r + z*s + w*s + t*u - u*s, x*t + x*a + x*b + y*s - y*a + z*r - w*r + w*s + v*s - v*a + v*b, x*t - x*s - w*t + t*u + u*r + v*r, x*r - y*r - z*r + z*a + t*u + t*v, x*t - x*a + x*b + y*r - y*s + w*t - w*r - t*u + u*r - u*s, x*t + x*s + x*b + z*t + z*s + z*a - z*b - u*r, t^2 - t*s + r^2 + r*a - 2*a^2 + a*b, x*t + x*r - x*a + y*s + z*r + w*a - w*b + t*v + u*r + u*s - v*a, x^2 - x*y - x*u - x*v - z^2 + t^2 + t*a - u*v - v^2, 2*t^2 + t*s - r^2 + r*a - r*b + s^2, x^2 - x*y - y*u - y*v - z^2 + z*v - w*u - w*v - t^2 + t*a - r^2 + r*a, x*z + y*z - y*u + z^2 + z*v - w*u - t^2 - t*r + r^2 + s*a, x^2 - x*y + x*z + x*w + x*v + y*z + y*w + z*w - z*u - t^2 + t*r - t*s - t*a - u*v + r*a - s*a, x^2 - x*y + x*z - x*w + x*u - y*z - y*w - y*v + z^2 + z*w - w*v + t*a + u*v - r^2 + r*a - s^2, 2*x*w - 2*x*u - z*w + z*u + w*v - t*r + t*s - t*a - u*v - r*b + s^2, x*u + y*u - z*u - w*u - t^2 + t*r - t*b - 2*u*v + s^2 - s*a, x*w + x*u + y*w + y*u + z*w - z*u + w*u - t*r - t*s - t*a + r^2 + r*b - s^2 - a^2, x^2 - x*y + x*w - x*u + y*w - y*u - y*v - z^2 - z*w - z*u + z*v - w^2 + w*u - t*r - t*s - r*s + r*a + s*a - a^2, 2*x*u - 2*y*u + 3*z*u + t*b - r*b - s*b + a*b, x*z + x*w - x*u - x*v + y*z + y*w + y*u - y*v - z^2 - z*u - 2*w*u + w*v - 2*t*s + 7*u^2 + 2*u*v - r*s - r*b - s^2 + s*a - s*b + a^2 + a*b - b^2]

Singular plane model Singular plane model

$ 0 $

$=$

$ 49 x^{8} y^{5} + 49 x^{8} y^{4} z - 49 x^{8} y^{3} z^{2} - 49 x^{8} y^{2} z^{3} - 266 x^{6} y^{6} z + \cdots - 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
$(-1/6:5/6:2/3:-2:0:-4/3:1:0:0:0:0)$, $(1:1:0:0:0:0:0:0:0:0:0)$, $(-1/2:1/2:0:0:0:0:1:0:0:0:0)$, $(-1/4:3/4:-1/2:1:0:0:0:0:0:0:0)$

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 $X_0(56)$ :

$\displaystyle X$	$=$	$\displaystyle -t$
$\displaystyle Y$	$=$	$\displaystyle -r$
$\displaystyle Z$	$=$	$\displaystyle -s$
$\displaystyle W$	$=$	$\displaystyle a$
$\displaystyle T$	$=$	$\displaystyle -b$

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.1 :

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

Equation of the image curve:

$0$

$=$

$ 49X^{8}Y^{5}-1806X^{4}Y^{9}+1344X^{2}Y^{11}+Y^{13}+49X^{8}Y^{4}Z-266X^{6}Y^{6}Z-2814X^{4}Y^{8}Z-3482X^{2}Y^{10}Z+17Y^{12}Z-49X^{8}Y^{3}Z^{2}-294X^{6}Y^{5}Z^{2}-5233X^{4}Y^{7}Z^{2}+3962X^{2}Y^{9}Z^{2}+23Y^{11}Z^{2}-49X^{8}Y^{2}Z^{3}+252X^{6}Y^{4}Z^{3}-2083X^{4}Y^{6}Z^{3}-3676X^{2}Y^{8}Z^{3}-375Y^{10}Z^{3}+196X^{6}Y^{3}Z^{4}-5267X^{4}Y^{5}Z^{4}+2454X^{2}Y^{7}Z^{4}+956Y^{9}Z^{4}+14X^{6}Y^{2}Z^{5}-1025X^{4}Y^{4}Z^{5}-526X^{2}Y^{6}Z^{5}-1246Y^{8}Z^{5}+98X^{6}YZ^{6}-1919X^{4}Y^{3}Z^{6}-318X^{2}Y^{5}Z^{6}+1071Y^{7}Z^{6}-365X^{4}Y^{2}Z^{7}+646X^{2}Y^{4}Z^{7}-665Y^{6}Z^{7}-175X^{4}YZ^{8}-462X^{2}Y^{3}Z^{8}+266Y^{5}Z^{8}-49X^{4}Z^{9}+112X^{2}Y^{2}Z^{9}-46Y^{4}Z^{9}-68X^{2}YZ^{10}-10Y^{3}Z^{10}+14X^{2}Z^{11}+12Y^{2}Z^{11}-3YZ^{12}-Z^{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.1.6	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

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