LMFDB - Modular curve 248.384.14-62.a.1.6

Invariants

Level:	$248$	$\SL_2$-level:	$124$	Newform level:	$124$
Index:	$384$	$\PSL_2$-index:	$192$
Genus:	$14 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (all of which are rational)	Cusp widths	$2^{3}\cdot62^{3}$	Cusp orbits	$1^{6}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 14$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 14$
Rational cusps:	$6$
Rational CM points:	none

Level structure

$\GL_2(\Z/248\Z)$-generators:	$\begin{bmatrix}5&90\\38&201\end{bmatrix}$, $\begin{bmatrix}17&64\\70&147\end{bmatrix}$, $\begin{bmatrix}33&92\\202&81\end{bmatrix}$, $\begin{bmatrix}123&110\\2&139\end{bmatrix}$, $\begin{bmatrix}127&166\\198&159\end{bmatrix}$, $\begin{bmatrix}153&236\\146&1\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 62.192.14.a.1 for the level structure with $-I$)
Cyclic 248-isogeny field degree:	$4$
Cyclic 248-torsion field degree:	$480$
Full 248-torsion field degree:	$3571200$

Models

Canonical model in $\mathbb{P}^{ 13 }$ defined by 66 equations

$ 0 $	$=$	$ y c + z c - b c - e^{2} $
	$=$	$x v + x r + x c + y^{2} - y b + y d - d e - e^{2}$
	$=$	$x z + x c + x e + y^{2} - y b + b c - c e$
	$=$	$x^{2} - x y - x z + x v + x r + x b - x d + x e - y b + y c - e^{2}$
	$=$	$\cdots$

[y*c + z*c - b*c - e^2, x*v + x*r + x*c + y^2 - y*b + y*d - d*e - e^2, x*z + x*c + x*e + y^2 - y*b + b*c - c*e, x^2 - x*y - x*z + x*v + x*r + x*b - x*d + x*e - y*b + y*c - e^2, x*y - x*c + x*d + x*e - y^2 - y*d + b*c - b*d + c*d, x*y + x*v + x*r - x*b + x*d + u*b - v*b, x^2 + x*e + y^2 - y*b + v*b + r*b - b*d - 2*e^2, x*z + x*c + y^2 - y*b + u*b - v*b + v*e + r*e - e^2, u*e - v*e - c*d - c*e + d*e, x*e + z*c - v*c - r*c + b*d + e^2, y*c - u*c + v*c - v*e - r*e, x^2 - x*y + x*t - x*v - x*r - x*a - y*s + z*u + s*c - c*d + d*e, x*y + x*t + x*s + x*c - x*e - y*a - y*c + z*u - u*e + e^2, x^2 - x*y - x*z + 2*x*u - x*v + x*r - x*c + x*d - b*d - e^2, x^2 - x*y - x*z + x*r - x*c + x*e + y*u + y*d + r*e - s*c + s*e - b*c - d*e - e^2, x*u + x*r - x*b - y*u - z*u - u*b + v*b - r*b - s*b + b*c + c*d - d*e, x*y + x*z - x*u - x*r - x*b - y*d + v*d + r*d - s*b + a*e + b*e + d*e, x^2 - x*z + 2*x*t - u*b + v*b + v*d + r*d + b*c - b*d - c*e, x*u + x*r - x*b + x*d + x*e - z*u - v*r - r^2 + r*d + r*e + s*e + b*c - c*e - d*e + e^2, x^2 - x*y - 2*x*w - x*e + u*e - v*c - r*c + a*c + c*e - e^2, x*u - x*e + y^2 + y*s - w*v - w*r + t*u + v*d + r*a - r*e + s^2 - s*d - s*e - e^2, x*r + x*s - x*a + z*u - v*r - v*e + r*s - r*c + s^2 + s*c - s*d + b*d - c*d + e^2, x*y + x*t + x*r - x*b + x*d - x*e + y*u - y*a + y*d - w*r + w*s + v*d + r*s - r*e - s*a - s*c + a*c - b*d - c*e - e^2, x*v - x*d - y*u + y*a - w*v + t*u - u*v - v*r + v*e + r*e + s*a + s*c - a*c + b*d + c*e + e^2, x*c - x*e - y*a + u*b + v*a + r*a - s*e + a*c - a*d - a*e - b*c - b*e - c*d + d*e - e^2, x*u + x*s + x*b + z*u - w*v + t*u - v*s - v*b - v*e - r*s + s*a + s*b + s*e + e^2, x^2 + x*t - x*u + x*e + y*u + z*u + u*r + v*s + r^2 + r*s - c*d - c*e + d*e, x*b + x*e - y*b - t*e - v*d - r*d + b*c + c*d - d*e, x*u - x*v + x*d + x*e - y*c + t*c + r*e - a*c - c*e + e^2, x*y - x*u + x*v + x*e + t*b + v*c - v*e + r*c - a*c - c*e, z*c + w*e + r*e - a*c - b*c, x*z - x*b + x*c - x*e + y^2 + y*c - y*d + u*d + v*e + r*d + a*c - e^2, x^2 - x*y - x*w + x*u + x*r + x*s - x*a + z*u + t*u - u*a - u*b - r*b + r*e + s^2 - s*d + s*e - d*e - e^2, x*w + x*t - x*v - x*r - x*d - x*e - y*s + u*s - v*s + v*d + v*e - s*e + d*e, x*u - x*v - x*b + x*d + y*b - z*u + v^2 + v*r + v*c - v*d - v*e + r*c + s*e - b*c - d*e, x*w - x*d - y*u + y*c - y*d - w*u + w*v - u*r + v*r - v*a - r*a - c*d - c*e + 2*d*e, x*b + x*e - y*b - y*d - u^2 + u*v - u*r + u*e + v*r - v*d - v*e - r*d + b*c + c*d, x^2 - x*c + x*d + x*e - y*d + z*u - w*v + t*u + t*d + v*r - v*e + r^2 + s*a - a*d - b*d + d*e, x*y + x*t - x*u + x*v - x*b + x*c + w*v + w*r - t*u - t*r + b*c - c*e + e^2, x*u + x*r + x*s - x*e - y*u - y*d + w*r + t*s + v*r - v*d + r^2 + r*s - r*a - r*d - b*d + c*d + c*e, x*z - x*u + x*v - x*r - x*s - x*b + x*c + y^2 + z*c + t*v + v*r + v*s - v*a - v*c + v*e - e^2, x*z - x*w - x*u - x*r - x*b + x*c - x*e + y*b - y*d + w*v + w*r - w*d + v*d + r*d - c*d - c*e + d*e + e^2, x*u - x*v - x*c + x*d + x*e + w*c + v*c - v*d - v*e + r*c - r*d - a*c - c*e + e^2, x*e + w*b + v*c - v*d - v*e + r*c - r*d - a*c - c*e, x*z + x*c + y^2 - y*b + z*e - e^2, x*b + x*d + x*e - y*b - z*d - e^2, x*e + z*b - u*b - r*b + b*c + c*d - d*e, x^2 - x*y + x*t - x*u - x*r - x*s - x*c - x*d + y*c - w*v + w*s + t*u + t*s + u*e - v*a - r*d + d*e - e^2, x*y - x*u + x*v + x*a + x*e + y*c - z*a + r*b - r*e + s*b, x*u + x*s + x*b + x*c + y^2 - y*u - y*b - z*s - e^2, x*u + x*r - x*b - z*u - z*r + b*c - c*e + e^2, x*u - x*v - x*b - x*e + y*b - y*c - z*u + z*v + e^2, x*y - x*z + x*t - x*b - x*c + x*d - y^2 + y*b - z*t + e^2, x*w + x*u - x*v - x*c + x*d + x*e - z*w, x*c - x*e + y^2 - y*c + z^2, x*z - x*b + x*c + y^2 - y*e, x*y - x*u - x*r - x*c + x*e - y^2 + y*u + y*r, x^2 - x*y - x*z + x*u + x*r + y*u - y*v + y*d - b*c - c*d - e^2, x*t - x*u + x*v + x*c - x*d - x*e - y*t, x*w + x*u - x*v - y*w - c*d - c*e + d*e, x*v + x*r + x*c - x*d - x*e - y*z, x^2 - x*y + x*r - x*b + x*e + y^2 + y*d + w*v + t*s - u*s - v*a + s*e - a*c - c*d - c*e, x*w + x*t + x*u - x*v + y^2 - y*c + y*d + w*s + t^2 + t*s - t*a - v*s + r*b - r*d - r*e + s*b - a*c + e^2, x*b - x*d + x*e - y^2 - y*a - y*b + w^2 - w*t + t^2 - v*r - r^2 - r*s + r*a - r*b + r*e - s*b + b*d - e^2, x*w - x*c - x*d + y*c - w*t - w*v - w*r + w*a + t*u + t*s - v*a + v*d - r*e - s*e - c*d - c*e + 2*d*e, x*y + x*w + x*t - x*u + x*a - y*c + z*u + t*a + r*a + r*e + s*a + s*e - a^2 - a*b - a*c + e^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
$(0:0:0:0:0:0:0:0:0:0:1:0:0:0)$, $(0:0:0:0:0:-1:-1:1:0:0:0:1:0:0)$, $(0:0:0:0:0:0:0:0:1:0:0:0:1:0)$, $(0:0:0:0:0:0:0:0:0:-1:1:0:0:0)$, $(0:0:0:0:0:0:0:0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:0:0:0:0:0:1:0:0)$

Maps to other modular curves

Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 62.64.4.a.1 :

$\displaystyle X$	$=$	$\displaystyle e$
$\displaystyle Y$	$=$	$\displaystyle -x$
$\displaystyle Z$	$=$	$\displaystyle -x-t-r-s+a$
$\displaystyle W$	$=$	$\displaystyle 2x-z+t+v+2r+s-a-b-c-d$

Equation of the image curve:

$0$	$=$	$ X^{2}-XY+Y^{2}+ZW $
	$=$	$ 2X^{3}+2X^{2}Y-XY^{2}-2Y^{3}+YZ^{2}+XZW-YZW+YW^{2} $

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	$8$	$32$	$32$	$0$	$0$
$X_0(31)$	$31$	$12$	$6$	$2$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
8.12.0-2.a.1.1	$8$	$32$	$32$	$0$	$0$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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