LMFDB - Modular curve 232.360.13-58.a.1.3

Invariants

Level:	$232$	$\SL_2$-level:	$116$	Newform level:	$116$
Index:	$360$	$\PSL_2$-index:	$180$
Genus:	$13 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (all of which are rational)	Cusp widths	$2^{3}\cdot58^{3}$	Cusp orbits	$1^{6}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 13$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 13$
Rational cusps:	$6$
Rational CM points:	none

Level structure

$\GL_2(\Z/232\Z)$-generators:	$\begin{bmatrix}57&174\\174&115\end{bmatrix}$, $\begin{bmatrix}71&24\\42&89\end{bmatrix}$, $\begin{bmatrix}91&24\\90&157\end{bmatrix}$, $\begin{bmatrix}119&40\\120&25\end{bmatrix}$, $\begin{bmatrix}167&68\\120&219\end{bmatrix}$, $\begin{bmatrix}211&150\\114&117\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 58.180.13.a.1 for the level structure with $-I$)
Cyclic 232-isogeny field degree:	$4$
Cyclic 232-torsion field degree:	$448$
Full 232-torsion field degree:	$2910208$

Models

Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations

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

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

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

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 $X_0(58)$ :

$\displaystyle X$	$=$	$\displaystyle v-d$
$\displaystyle Y$	$=$	$\displaystyle y+d$
$\displaystyle Z$	$=$	$\displaystyle y-a+b+d$
$\displaystyle W$	$=$	$\displaystyle x+y+w+u-r-d$
$\displaystyle T$	$=$	$\displaystyle -x-y-t-u+r+d$
$\displaystyle U$	$=$	$\displaystyle c$

Equation of the image curve:

$0$	$=$	$ Y^{2}-YZ-YU-TU $
	$=$	$ XY+Y^{2}-XW-WT-YU $
	$=$	$ XY+YZ-XW-YW+ZW+W^{2}+YU $
	$=$	$ 2XZ+YZ+XW+W^{2}-XT-YT-T^{2}+YU-2WU $
	$=$	$ XY-2XZ-YZ-ZW-W^{2}-XT-ZT-2XU-YU+ZU+WU $
	$=$	$ 2X^{2}+2XY+XZ+YZ+XW+YW+W^{2}+XT+YT+WT-2WU $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
8.12.0-2.a.1.1	$8$	$30$	$30$	$0$	$0$
$X_0(29)$	$29$	$12$	$6$	$2$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
8.12.0-2.a.1.1	$8$	$30$	$30$	$0$	$0$

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Elliptic curves over $\Q$
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