LMFDB - Modular curve 112.384.11-56.fb.2.25

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:	$\begin{bmatrix}11&32\\62&37\end{bmatrix}$, $\begin{bmatrix}33&6\\98&109\end{bmatrix}$, $\begin{bmatrix}42&3\\47&110\end{bmatrix}$, $\begin{bmatrix}65&70\\26&53\end{bmatrix}$, $\begin{bmatrix}80&31\\29&26\end{bmatrix}$, $\begin{bmatrix}90&75\\59&106\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fb.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 $	$=$	$ z r - t a $
	$=$	$y r + z t$
	$=$	$t s - t b + u r$
	$=$	$y s - y b - z u$
	$=$	$\cdots$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ - x^{14} y^{5} z + 2 x^{14} y^{4} z^{2} + x^{12} y^{8} - 22 x^{12} y^{7} z + 10 x^{12} y^{6} z^{2} + \cdots + 331776 z^{20} $

[-x^14*y^5*z + 2*x^14*y^4*z^2 + x^12*y^8 - 22*x^12*y^7*z + 10*x^12*y^6*z^2 + 42*x^12*y^5*z^3 + 34*x^12*y^4*z^4 + 20*x^12*y^3*z^5 - 27*x^12*y^2*z^6 + 54*x^12*y*z^7 + 20*x^10*y^10 - 80*x^10*y^9*z - 212*x^10*y^8*z^2 + 172*x^10*y^7*z^3 + 444*x^10*y^6*z^4 + 916*x^10*y^5*z^5 + 200*x^10*y^4*z^6 + 580*x^10*y^3*z^7 + 1116*x^10*y^2*z^8 + 1188*x^10*y*z^9 + 216*x^10*z^10 + 64*x^8*y^12 + 64*x^8*y^11*z - 464*x^8*y^10*z^2 - 704*x^8*y^9*z^3 - 588*x^8*y^8*z^4 + 2360*x^8*y^7*z^5 + 5732*x^8*y^6*z^6 + 4448*x^8*y^5*z^7 + 12220*x^8*y^4*z^8 + 20440*x^8*y^3*z^9 + 24412*x^8*y^2*z^10 + 14800*x^8*y*z^11 + 3456*x^8*z^12 - 1024*x^6*y^10*z^4 - 2624*x^6*y^9*z^5 + 1328*x^6*y^8*z^6 + 2288*x^6*y^7*z^7 + 13856*x^6*y^6*z^8 + 57936*x^6*y^5*z^9 + 158640*x^6*y^4*z^10 + 229056*x^6*y^3*z^11 + 214368*x^6*y^2*z^12 + 118496*x^6*y*z^13 + 26560*x^6*z^14 - 18304*x^4*y^8*z^8 - 36416*x^4*y^7*z^9 + 50288*x^4*y^6*z^10 + 316000*x^4*y^5*z^11 + 746976*x^4*y^4*z^12 + 1068608*x^4*y^3*z^13 + 922160*x^4*y^2*z^14 + 527456*x^4*y*z^15 + 128000*x^4*z^16 + 64512*x^2*y^6*z^12 + 536256*x^2*y^5*z^13 + 1198656*x^2*y^4*z^14 + 1881792*x^2*y^3*z^15 + 1752768*x^2*y^2*z^16 + 1078848*x^2*y*z^17 + 339840*x^2*z^18 + 254016*y^4*z^16 + 508032*y^3*z^17 + 834624*y^2*z^18 + 580608*y*z^19 + 331776*z^20]

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

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 $X_0(56)$ :

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

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.fb.2 :

$\displaystyle X$	$=$	$\displaystyle a$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{4}b$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{2}r$

Equation of the image curve:

$0$

$=$

$ X^{12}Y^{8}+20X^{10}Y^{10}+64X^{8}Y^{12}-X^{14}Y^{5}Z-22X^{12}Y^{7}Z-80X^{10}Y^{9}Z+64X^{8}Y^{11}Z+2X^{14}Y^{4}Z^{2}+10X^{12}Y^{6}Z^{2}-212X^{10}Y^{8}Z^{2}-464X^{8}Y^{10}Z^{2}+42X^{12}Y^{5}Z^{3}+172X^{10}Y^{7}Z^{3}-704X^{8}Y^{9}Z^{3}+34X^{12}Y^{4}Z^{4}+444X^{10}Y^{6}Z^{4}-588X^{8}Y^{8}Z^{4}-1024X^{6}Y^{10}Z^{4}+20X^{12}Y^{3}Z^{5}+916X^{10}Y^{5}Z^{5}+2360X^{8}Y^{7}Z^{5}-2624X^{6}Y^{9}Z^{5}-27X^{12}Y^{2}Z^{6}+200X^{10}Y^{4}Z^{6}+5732X^{8}Y^{6}Z^{6}+1328X^{6}Y^{8}Z^{6}+54X^{12}YZ^{7}+580X^{10}Y^{3}Z^{7}+4448X^{8}Y^{5}Z^{7}+2288X^{6}Y^{7}Z^{7}+1116X^{10}Y^{2}Z^{8}+12220X^{8}Y^{4}Z^{8}+13856X^{6}Y^{6}Z^{8}-18304X^{4}Y^{8}Z^{8}+1188X^{10}YZ^{9}+20440X^{8}Y^{3}Z^{9}+57936X^{6}Y^{5}Z^{9}-36416X^{4}Y^{7}Z^{9}+216X^{10}Z^{10}+24412X^{8}Y^{2}Z^{10}+158640X^{6}Y^{4}Z^{10}+50288X^{4}Y^{6}Z^{10}+14800X^{8}YZ^{11}+229056X^{6}Y^{3}Z^{11}+316000X^{4}Y^{5}Z^{11}+3456X^{8}Z^{12}+214368X^{6}Y^{2}Z^{12}+746976X^{4}Y^{4}Z^{12}+64512X^{2}Y^{6}Z^{12}+118496X^{6}YZ^{13}+1068608X^{4}Y^{3}Z^{13}+536256X^{2}Y^{5}Z^{13}+26560X^{6}Z^{14}+922160X^{4}Y^{2}Z^{14}+1198656X^{2}Y^{4}Z^{14}+527456X^{4}YZ^{15}+1881792X^{2}Y^{3}Z^{15}+128000X^{4}Z^{16}+1752768X^{2}Y^{2}Z^{16}+254016Y^{4}Z^{16}+1078848X^{2}YZ^{17}+508032Y^{3}Z^{17}+339840X^{2}Z^{18}+834624Y^{2}Z^{18}+580608YZ^{19}+331776Z^{20} $

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.1.7	$16$	$8$	$8$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
16.48.0-8.ba.1.7	$16$	$8$	$8$	$0$	$0$
112.192.5-56.bl.1.20	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

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