LMFDB - Modular curve 168.384.11-56.v.1.5

Invariants

Level:	$168$	$\SL_2$-level:	$28$	Newform level:	$448$
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 $2$ are rational)	Cusp widths	$4^{6}\cdot28^{6}$	Cusp orbits	$1^{2}\cdot2^{5}$
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:	$2$
Rational CM points:	none

Level structure

$\GL_2(\Z/168\Z)$-generators:	$\begin{bmatrix}31&158\\76&103\end{bmatrix}$, $\begin{bmatrix}47&116\\50&51\end{bmatrix}$, $\begin{bmatrix}75&22\\92&117\end{bmatrix}$, $\begin{bmatrix}75&50\\22&61\end{bmatrix}$, $\begin{bmatrix}77&144\\158&161\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.v.1 for the level structure with $-I$)
Cyclic 168-isogeny field degree:	$16$
Cyclic 168-torsion field degree:	$768$
Full 168-torsion field degree:	$387072$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

$ 0 $	$=$	$ x z + x s + y v - y a - u b $
	$=$	$x b + y r + z w + w s$
	$=$	$x y - x b + 2 y^{2} - z^{2} - z w - t s$
	$=$	$x b + z t - w t + t v + t s + u s + v s - r b$
	$=$	$\cdots$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 24 x^{6} y^{8} - 112 x^{6} y^{7} z + 128 x^{6} y^{6} z^{2} + 112 x^{6} y^{5} z^{3} - 304 x^{6} y^{4} z^{4} + \cdots + 32 y z^{13} $

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:1/2:1/2:0:1/2:0:0:1:0)$, $(0:0:1:-1:0:0:0:0:-1:1:0)$

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 28.96.4.a.1 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle -y$
$\displaystyle Z$	$=$	$\displaystyle -r$
$\displaystyle W$	$=$	$\displaystyle b$

Equation of the image curve:

$0$	$=$	$ 7X^{2}+Z^{2}-2ZW $
	$=$	$ 21XY^{2}+14Y^{3}-4YZ^{2}+4XZW+8YZW-XW^{2} $

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

$\displaystyle X$	$=$	$\displaystyle b$
$\displaystyle Y$	$=$	$\displaystyle 2z$
$\displaystyle Z$	$=$	$\displaystyle 2t$

Equation of the image curve:

$0$

$=$

$ 24X^{6}Y^{8}+8X^{4}Y^{10}+16X^{2}Y^{12}-112X^{6}Y^{7}Z-12X^{4}Y^{9}Z-104X^{2}Y^{11}Z+32Y^{13}Z+128X^{6}Y^{6}Z^{2}-24X^{4}Y^{8}Z^{2}+284X^{2}Y^{10}Z^{2}-240Y^{12}Z^{2}+112X^{6}Y^{5}Z^{3}+48X^{4}Y^{7}Z^{3}-374X^{2}Y^{9}Z^{3}+912Y^{11}Z^{3}-304X^{6}Y^{4}Z^{4}+16X^{4}Y^{6}Z^{4}+92X^{2}Y^{8}Z^{4}-2280Y^{10}Z^{4}+112X^{6}Y^{3}Z^{5}-72X^{4}Y^{5}Z^{5}+478X^{2}Y^{7}Z^{5}+4170Y^{9}Z^{5}+128X^{6}Y^{2}Z^{6}+16X^{4}Y^{4}Z^{6}-784X^{2}Y^{6}Z^{6}-5883Y^{8}Z^{6}-112X^{6}YZ^{7}+48X^{4}Y^{3}Z^{7}+478X^{2}Y^{5}Z^{7}+6580Y^{7}Z^{7}+24X^{6}Z^{8}-24X^{4}Y^{2}Z^{8}+92X^{2}Y^{4}Z^{8}-5883Y^{6}Z^{8}-12X^{4}YZ^{9}-374X^{2}Y^{3}Z^{9}+4170Y^{5}Z^{9}+8X^{4}Z^{10}+284X^{2}Y^{2}Z^{10}-2280Y^{4}Z^{10}-104X^{2}YZ^{11}+912Y^{3}Z^{11}+16X^{2}Z^{12}-240Y^{2}Z^{12}+32YZ^{13} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
84.192.4-28.a.1.5	$84$	$2$	$2$	$4$	$?$
168.192.4-28.a.1.12	$168$	$2$	$2$	$4$	$?$
168.192.5-56.a.1.1	$168$	$2$	$2$	$5$	$?$
168.192.5-56.a.1.14	$168$	$2$	$2$	$5$	$?$
168.192.6-56.a.1.1	$168$	$2$	$2$	$6$	$?$
168.192.6-56.a.1.10	$168$	$2$	$2$	$6$	$?$

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