LMFDB - Modular curve 112.384.11-56.ff.3.28

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:	$\begin{bmatrix}8&105\\33&80\end{bmatrix}$, $\begin{bmatrix}15&36\\66&41\end{bmatrix}$, $\begin{bmatrix}29&38\\18&105\end{bmatrix}$, $\begin{bmatrix}31&56\\30&1\end{bmatrix}$, $\begin{bmatrix}91&4\\16&23\end{bmatrix}$, $\begin{bmatrix}106&35\\5&24\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.ff.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 t + x a + y t + z s + u v $
	$=$	$x t - x s - y t - w s$
	$=$	$x v + x b - y t - y v + w r + w a + w b$
	$=$	$x v - x r - x b + y t + y v - w v$
	$=$	$\cdots$

[x*t + x*a + y*t + z*s + u*v, x*t - x*s - y*t - w*s, x*v + x*b - y*t - y*v + w*r + w*a + w*b, x*v - x*r - x*b + y*t + y*v - w*v, x*a - y*v + y*r + w*t - w*v, x*v - x*a - y*t + y*s - z*v, x*v - x*r + x*s - z*t - z*v, x*t + x*s + y*v - z*r + w*v + w*s + w*a, x*t + x*r - y*t + z*t - z*a + w*t + t*u, x*w + x*u + y*z, y*v - y*r + y*s + w*t + w*v + t*u + u*v, x*t - x*v + x*s - y*v + y*r + y*a + y*b - z*v, x*r + x*s + x*a + x*b + y*s - z*r + z*s - z*b - w*t - t*u, x*s - x*a - y*v - y*a - z*t - w*v + w*s, y*b + z*r + z*a - w*t - w*r - w*s + w*b + u*v - u*r, x*r + y*r + y*s + y*a - z*a - w*s - u*v + u*r - u*s + u*a + u*b, x*v - x*r - x*s - y*a - y*b + z*t - z*v - z*a + w*t + w*r + w*a - w*b + u*v - u*b, x^2 + x*z + y^2 + y*w - 2*w^2 - w*u, 2*x^2 - x*z - y^2 + y*w + y*u + z^2, 2*x*y + x*w - 2*x*u + y*z + 2*z*w + z*u + v*r - v*a - r^2 - r*s - r*b - s*a + a^2 + a*b, x^2 - 2*x*z + y^2 + z^2 - w^2 + 2*u^2 - v*r + 2*v*a + r^2 + 2*r*s + r*a + 2*r*b + 2*s*a - a*b, t^2 + t*s - 3*v^2 - v*a - 2*v*b + r^2 + r*a + 2*r*b + a*b + b^2, t*v + t*s + t*a + t*b + v*r - 3*v*s - v*a + r*s + s^2 + s*b - a^2, 2*x*y + x*w - 2*x*u + y*z + 2*z*w + z*u + t^2 - t*v - 2*t*s - t*b - 2*v*r + v*a - r*a - s^2 + s*a - a*b, 2*t*v - t*r - t*s - t*a - 2*t*b - 2*v*r + 2*v*a + r^2 + r*s + r*b - s^2 + s*a - a*b, x*y - x*w + x*u - y*z + z*w - 3*z*u - t*r - t*a + v*r + a*b, 2*x^2 - 2*x*z + y^2 - y*u - w^2 + w*u + t^2 - t*v - 2*t*s - t*b - u^2 - v*r - r*s - r*a - r*b - s^2, x^2 - 2*x*z - 2*y*u + z^2 - 2*w*u + t^2 + t*s - u^2 + 2*v*r - v*a - r*s + r*a - s*a + a*b, 2*x^2 - 2*y*u - 2*z^2 + 2*w*u + u^2 + r*s + s*a, x^2 + x*z - 2*y^2 - 3*y*w - z^2 - 2*w^2 + w*u - t^2 + t*v + t*s + t*a + t*b + u^2 + r*s + s^2, 2*x*y - x*w + 3*x*u - y*z + 2*z*w + z*u - t*v + t*r + t*a + t*b - 2*v*r + r^2 + r*s + r*b + s*a - a^2 - a*b, t^2 + t*v + t*r + 2*t*s + 2*t*a + t*b + 3*v*s + v*a + r^2 - 2*r*s + r*a + r*b + s^2 - s*a - s*b, x*z + y*w - y*u + z^2 - w^2 + 3*w*u + t^2 + t*v - t*s + t*a - t*b - 2*v*r + r^2 - r*s, x^2 - 4*x*y + 3*x*z - x*u - 2*y*w - z^2 + 3*z*w - 2*z*u + 2*w^2 - w*u + t^2 - t*r - u^2 + 3*v^2 - v*r - v*b - r*s - r*a - a^2, 2*x^2 + 5*x*w - 2*x*u - 2*y*z - 2*y*w + y*u + z^2 + 2*w^2 - w*u - t*s - u^2 + 3*v^2 - v*r - v*a - v*b - r^2 - r*s - r*a - r*b - s*a, 3*x^2 + 5*x*y - 3*x*w + 2*y^2 + 4*y*z - 4*y*w + 3*z^2 - 2*z*w - z*u + 2*w^2 + w*u - v*s + s*b]

Singular plane model Singular plane model

$ 0 $

$=$

$ 6272 x^{13} y^{4} z^{2} + 43904 x^{12} y^{6} z + 288512 x^{12} y^{4} z^{3} + 5376 x^{12} y^{2} z^{5} + \cdots - 1094014992384 z^{19} $

[6272*x^13*y^4*z^2 + 43904*x^12*y^6*z + 288512*x^12*y^4*z^3 + 5376*x^12*y^2*z^5 + 76832*x^11*y^8 + 2129344*x^11*y^6*z^2 + 6325312*x^11*y^4*z^4 + 279552*x^11*y^2*z^6 - 2304960*x^10*y^8*z + 45349696*x^10*y^6*z^3 + 60264512*x^10*y^4*z^5 + 7066752*x^10*y^2*z^7 - 8605184*x^9*y^10 - 105550704*x^9*y^8*z^2 + 47000800*x^9*y^6*z^4 + 125948872*x^9*y^4*z^6 + 88975488*x^9*y^2*z^8 - 221184*x^9*z^10 + 203259056*x^8*y^10*z - 309331120*x^8*y^8*z^3 + 2713816*x^8*y^6*z^5 - 2115721888*x^8*y^4*z^7 + 466607568*x^8*y^2*z^9 - 13934592*x^8*z^11 + 361417728*x^7*y^12 - 5431907152*x^7*y^10*z^2 + 18855900210*x^7*y^8*z^4 + 48317372572*x^7*y^6*z^6 + 10811879344*x^7*y^4*z^8 - 1793016288*x^7*y^2*z^10 - 433926144*x^7*z^12 - 28186280192*x^6*y^12*z - 455948651480*x^6*y^10*z^3 - 1169889895216*x^6*y^8*z^5 - 287844580016*x^6*y^6*z^7 + 178829647192*x^6*y^4*z^9 - 17482018240*x^6*y^2*z^11 - 7823130624*x^6*z^13 - 6746464256*x^5*y^14 + 397178721408*x^5*y^12*z^2 - 2908645053076*x^5*y^10*z^4 - 5021379519548*x^5*y^8*z^6 - 6750943531416*x^5*y^6*z^8 - 1358072693544*x^5*y^4*z^10 + 70485665440*x^5*y^2*z^12 - 86617241088*x^5*z^14 + 1320439669248*x^4*y^14*z + 16360457237208*x^4*y^12*z^3 + 94031052843315*x^4*y^10*z^5 + 131868768324930*x^4*y^8*z^7 + 78374394819504*x^4*y^6*z^9 + 7893543451136*x^4*y^4*z^11 - 380913544784*x^4*y^2*z^13 - 551476821504*x^4*z^15 + 47225249792*x^3*y^16 - 23183652171008*x^3*y^14*z^2 - 122929074380294*x^3*y^12*z^4 - 310469179406036*x^3*y^10*z^6 + 42118389484234*x^3*y^8*z^8 + 272661777162852*x^3*y^6*z^10 + 108736224898672*x^3*y^4*z^12 - 18838733048128*x^3*y^2*z^14 - 1835443126272*x^3*z^16 - 21436890173440*x^2*y^16*z + 200225760141120*x^2*y^14*z^3 + 86769431705614*x^2*y^12*z^5 - 1114739612266266*x^2*y^10*z^7 - 3131009644465108*x^2*y^8*z^9 - 2619622699377208*x^2*y^6*z^11 - 706269631902520*x^2*y^4*z^13 - 150417125053504*x^2*y^2*z^15 - 3256388966400*x^2*z^17 + 1057400328699904*x*y^16*z^2 + 14914803537727296*x*y^14*z^4 + 49408145176238380*x*y^12*z^6 + 73442550928362420*x*y^10*z^8 + 51937131984756928*x*y^8*z^10 + 13055426995115400*x*y^6*z^12 - 3094205089934112*x*y^4*z^14 - 324218453452800*x*y^2*z^16 - 2958420738048*x*z^18 + 161958994161664*y^18*z - 2421631011311232*y^16*z^3 - 89830259999422341*y^14*z^5 - 300038767764303232*y^12*z^7 - 422647972707167529*y^10*z^9 - 315870644119281006*y^8*z^11 - 117285455306645024*y^6*z^13 - 11465462957124960*y^4*z^15 - 195870847939584*y^2*z^17 - 1094014992384*z^19]

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
$(0:0:0:0:4/7:0:1/7:-4/7:0:-4/7:1)$, $(0:0:0:0:0:0:1:2:0:-2:1)$, $(0:0:0:0:0:0:1/3:0:0:0:1)$, $(0:0:0:0:-2:0:1:-2:-2:2:1)$

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 $X_0(56)$ :

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

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.3 :

$\displaystyle X$	$=$	$\displaystyle -\frac{7}{4}t+b$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{4}u$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{4}s$

Equation of the image curve:

$0$

$=$

$ 76832X^{11}Y^{8}-8605184X^{9}Y^{10}+361417728X^{7}Y^{12}-6746464256X^{5}Y^{14}+47225249792X^{3}Y^{16}+43904X^{12}Y^{6}Z-2304960X^{10}Y^{8}Z+203259056X^{8}Y^{10}Z-28186280192X^{6}Y^{12}Z+1320439669248X^{4}Y^{14}Z-21436890173440X^{2}Y^{16}Z+161958994161664Y^{18}Z+6272X^{13}Y^{4}Z^{2}+2129344X^{11}Y^{6}Z^{2}-105550704X^{9}Y^{8}Z^{2}-5431907152X^{7}Y^{10}Z^{2}+397178721408X^{5}Y^{12}Z^{2}-23183652171008X^{3}Y^{14}Z^{2}+1057400328699904XY^{16}Z^{2}+288512X^{12}Y^{4}Z^{3}+45349696X^{10}Y^{6}Z^{3}-309331120X^{8}Y^{8}Z^{3}-455948651480X^{6}Y^{10}Z^{3}+16360457237208X^{4}Y^{12}Z^{3}+200225760141120X^{2}Y^{14}Z^{3}-2421631011311232Y^{16}Z^{3}+6325312X^{11}Y^{4}Z^{4}+47000800X^{9}Y^{6}Z^{4}+18855900210X^{7}Y^{8}Z^{4}-2908645053076X^{5}Y^{10}Z^{4}-122929074380294X^{3}Y^{12}Z^{4}+14914803537727296XY^{14}Z^{4}+5376X^{12}Y^{2}Z^{5}+60264512X^{10}Y^{4}Z^{5}+2713816X^{8}Y^{6}Z^{5}-1169889895216X^{6}Y^{8}Z^{5}+94031052843315X^{4}Y^{10}Z^{5}+86769431705614X^{2}Y^{12}Z^{5}-89830259999422341Y^{14}Z^{5}+279552X^{11}Y^{2}Z^{6}+125948872X^{9}Y^{4}Z^{6}+48317372572X^{7}Y^{6}Z^{6}-5021379519548X^{5}Y^{8}Z^{6}-310469179406036X^{3}Y^{10}Z^{6}+49408145176238380XY^{12}Z^{6}+7066752X^{10}Y^{2}Z^{7}-2115721888X^{8}Y^{4}Z^{7}-287844580016X^{6}Y^{6}Z^{7}+131868768324930X^{4}Y^{8}Z^{7}-1114739612266266X^{2}Y^{10}Z^{7}-300038767764303232Y^{12}Z^{7}+88975488X^{9}Y^{2}Z^{8}+10811879344X^{7}Y^{4}Z^{8}-6750943531416X^{5}Y^{6}Z^{8}+42118389484234X^{3}Y^{8}Z^{8}+73442550928362420XY^{10}Z^{8}+466607568X^{8}Y^{2}Z^{9}+178829647192X^{6}Y^{4}Z^{9}+78374394819504X^{4}Y^{6}Z^{9}-3131009644465108X^{2}Y^{8}Z^{9}-422647972707167529Y^{10}Z^{9}-221184X^{9}Z^{10}-1793016288X^{7}Y^{2}Z^{10}-1358072693544X^{5}Y^{4}Z^{10}+272661777162852X^{3}Y^{6}Z^{10}+51937131984756928XY^{8}Z^{10}-13934592X^{8}Z^{11}-17482018240X^{6}Y^{2}Z^{11}+7893543451136X^{4}Y^{4}Z^{11}-2619622699377208X^{2}Y^{6}Z^{11}-315870644119281006Y^{8}Z^{11}-433926144X^{7}Z^{12}+70485665440X^{5}Y^{2}Z^{12}+108736224898672X^{3}Y^{4}Z^{12}+13055426995115400XY^{6}Z^{12}-7823130624X^{6}Z^{13}-380913544784X^{4}Y^{2}Z^{13}-706269631902520X^{2}Y^{4}Z^{13}-117285455306645024Y^{6}Z^{13}-86617241088X^{5}Z^{14}-18838733048128X^{3}Y^{2}Z^{14}-3094205089934112XY^{4}Z^{14}-551476821504X^{4}Z^{15}-150417125053504X^{2}Y^{2}Z^{15}-11465462957124960Y^{4}Z^{15}-1835443126272X^{3}Z^{16}-324218453452800XY^{2}Z^{16}-3256388966400X^{2}Z^{17}-195870847939584Y^{2}Z^{17}-2958420738048XZ^{18}-1094014992384Z^{19} $

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	$112$	$2$	$2$	$5$	$?$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi
Siegel

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Modular curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Galois groups
Sato-Tate groups
Abstract groups
Lattices

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Motives

Groups

Database

Properties

Related objects

Downloads

Learn more