LMFDB - Modular curve 112.384.11-56.fb.1.25

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:	$2 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 11$
Rational cusps:	$4$
Rational CM points:	none

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}14&47\\69&104\end{bmatrix}$, $\begin{bmatrix}39&78\\62&55\end{bmatrix}$, $\begin{bmatrix}43&2\\0&101\end{bmatrix}$, $\begin{bmatrix}89&92\\62&63\end{bmatrix}$, $\begin{bmatrix}90&17\\3&48\end{bmatrix}$, $\begin{bmatrix}111&110\\110&111\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fb.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 w - y v - z w + r a $
	$=$	$x y - y w - y r + t r$
	$=$	$x w - x s - w v$
	$=$	$x t + y b - w t$
	$=$	$\cdots$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 4 x^{8} y^{5} z^{2} + 16 x^{6} y^{9} - 64 x^{6} y^{8} z + 124 x^{6} y^{7} z^{2} - 140 x^{6} y^{6} z^{3} + \cdots + 4 y^{3} z^{12} $

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
$(0:0:0:0:0:0:4:1:0:0:0)$, $(1:0:0:0:0:0:1:1/2:0:0:1)$, $(0:0:0:0:0:0:0:1/4:1:0:0)$, $(0:0:0:-1:0:0:0:1/2:0:0:1)$

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 $X_0(56)$ :

$\displaystyle X$	$=$	$\displaystyle -y$
$\displaystyle Y$	$=$	$\displaystyle -z$
$\displaystyle Z$	$=$	$\displaystyle t$
$\displaystyle W$	$=$	$\displaystyle u$
$\displaystyle T$	$=$	$\displaystyle -a$

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.fb.1 :

$\displaystyle X$	$=$	$\displaystyle a$
$\displaystyle Y$	$=$	$\displaystyle w$
$\displaystyle Z$	$=$	$\displaystyle s$

Equation of the image curve:

$0$

$=$

$ 16X^{6}Y^{9}+16X^{4}Y^{11}-64X^{6}Y^{8}Z-64X^{4}Y^{10}Z+16X^{2}Y^{12}Z+4X^{8}Y^{5}Z^{2}+124X^{6}Y^{7}Z^{2}+156X^{4}Y^{9}Z^{2}-56X^{2}Y^{11}Z^{2}-140X^{6}Y^{6}Z^{3}-236X^{4}Y^{8}Z^{3}+132X^{2}Y^{10}Z^{3}+122X^{6}Y^{5}Z^{4}+242X^{4}Y^{7}Z^{4}-230X^{2}Y^{9}Z^{4}+36Y^{11}Z^{4}-68X^{6}Y^{4}Z^{5}-146X^{4}Y^{6}Z^{5}+370X^{2}Y^{8}Z^{5}-108Y^{10}Z^{5}+44X^{6}Y^{3}Z^{6}+71X^{4}Y^{5}Z^{6}-519X^{2}Y^{7}Z^{6}+165Y^{9}Z^{6}-16X^{6}Y^{2}Z^{7}-2X^{4}Y^{4}Z^{7}+623X^{2}Y^{6}Z^{7}-150Y^{8}Z^{7}+2X^{6}YZ^{8}-10X^{4}Y^{3}Z^{8}-539X^{2}Y^{5}Z^{8}+109Y^{7}Z^{8}+26X^{4}Y^{2}Z^{9}+385X^{2}Y^{4}Z^{9}-64Y^{6}Z^{9}-14X^{4}YZ^{10}-230X^{2}Y^{3}Z^{10}+32Y^{5}Z^{10}+2X^{4}Z^{11}+120X^{2}Y^{2}Z^{11}-8Y^{4}Z^{11}-36X^{2}YZ^{12}+4Y^{3}Z^{12}+4X^{2}Z^{13} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(7)$	$7$	$48$	$24$	$0$	$0$
16.48.0-8.ba.2.4	$16$	$8$	$8$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
16.48.0-8.ba.2.4	$16$	$8$	$8$	$0$	$0$
112.192.5-56.bl.1.23	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

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