LMFDB - Modular curve 120.384.13-60.bl.2.4

Invariants

Level:	$120$	$\SL_2$-level:	$60$	Newform level:	$720$
Index:	$384$	$\PSL_2$-index:	$192$
Genus:	$13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)	Cusp widths	$4^{2}\cdot12^{2}\cdot20^{2}\cdot60^{2}$	Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 24$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 13$
Rational cusps:	$0$
Rational CM points:	none

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}9&34\\71&67\end{bmatrix}$, $\begin{bmatrix}61&98\\114&35\end{bmatrix}$, $\begin{bmatrix}98&103\\77&9\end{bmatrix}$, $\begin{bmatrix}103&14\\57&5\end{bmatrix}$, $\begin{bmatrix}105&53\\83&30\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.192.13.bl.2 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$12$
Cyclic 120-torsion field degree:	$384$
Full 120-torsion field degree:	$92160$

Models

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

$ 0 $	$=$	$ x w + x t - x v + x a - 2 x b + x d + t r + t s $
	$=$	$x^{2} + x y + x z - x w + x u + x v + x r - x s - 3 x a - 2 z r + t s + v r$
	$=$	$x y - x z - x r + x s - x a - x d + y a - y d - z a - w a + w d - t r - t s + t a + t d - u a + u d - v a$
	$=$	$x y - x u - x v - x r + x s - 2 x a - x b - x c + y^{2} + y z + y w + y t - y v - y r - 3 y b + y c + \cdots + v c$
	$=$	$\cdots$

[x*w + x*t - x*v + x*a - 2*x*b + x*d + t*r + t*s, x^2 + x*y + x*z - x*w + x*u + x*v + x*r - x*s - 3*x*a - 2*z*r + t*s + v*r, x*y - x*z - x*r + x*s - x*a - x*d + y*a - y*d - z*a - w*a + w*d - t*r - t*s + t*a + t*d - u*a + u*d - v*a, x*y - x*u - x*v - x*r + x*s - 2*x*a - x*b - x*c + y^2 + y*z + y*w + y*t - y*v - y*r - 3*y*b + y*c - z^2 - z*r - u*s + u*c + v*r + v*c, x*y - x*z - 2*x*r - 2*x*a - y*a + y*d + z*r + z*s + 2*z*a - w*d - t*a - t*d + u*a - u*d + v*r + v*s + v*a, x*w - x*r - 3*x*s - x*a + 2*x*c - x*d + y*a - y*d - z*a - w*a - t*r - t*s - v*a - v*d, x^2 + x*y + x*z - x*w + x*u + x*v - 2*x*s + x*a + 2*x*d + y*a - y*d + 2*z*s - z*a - w*a - t*r - v*s - v*a - v*d, x*r + x*s + x*a - 2*y*r - 2*y*s - y*d + z*r + z*s - z*a + w*a + w*d + t*a + t*d - u*r - u*s + u*a + u*d - v*r - v*s, 2*x*r + 2*x*s - x*a - x*d - y*z + y*w + y*a - y*d + z^2 - z*w + z*r - z*s - 2*z*a - w*r + w*s + w*a + w*d + t*a + t*d, x*y - x*z + x*w - 2*x*a + 2*x*c + z*r + z*s - z*a - z*d + w*a + u*a - u*d + v*r + v*s - v*d, x*r + x*s - 2*x*a + y*a - y*d + z*r + z*s + 2*z*a + z*d + w*a + u*a - u*d + v*r + v*s + v*a, x*r + x*s - x*a + x*d + z*r + z*s + z*a + w*a + t*u - t*r - t*s - t*a - u*v + 2*u*s - 2*u*b - 2*u*c + v*r + v*s - v*a, x^2 + 2*x*y + 2*x*v + x*r - 2*x*s + x*b + 2*x*c + y^2 - y*z - y*w - y*t + y*u - y*r + y*s + y*a - y*d + z^2 - z*s - z*a - t*r + t*s - u*r + u*a + u*c - u*d - v*r - v*d, x*y + x*w - x*u - x*v - x*b + x*c - y^2 + 2*y*z - y*w + y*r + y*a - y*b - y*c - y*d + 2*z*r + z*s - z*a - w*s - w*a + w*b + w*c - t*s + t*c - u*s + u*c - v*a - v*d, 2*x^2 - x*r - x*s - 2*x*a + y^2 - y*z + y*w - y*v - y*r - y*b + y*c - z*r - w*v - w*b - t*u - t*v - t*r - t*a - u*v + v*r + v*b + v*c, 2*x*a + y*a - y*d - z*r - z*s - 2*z*a - z*d - w*t + w*a - w*d + 2*t*s - 2*t*c + u*a - u*d - v*r - v*s, x*w + x*t - x*u - x*s - 2*x*a - x*b + x*d + 2*y^2 - y*t + y*u - 2*y*r - y*a + 2*y*c + y*d - z^2 - z*r + z*a + w*v + w*a - w*d + t*s + t*a + t*d - u*r + u*a + u*c - u*d + v*a + 2*v*c, 3*x^2 + x*y - 2*x*z + x*w + x*t + x*v + x*s + x*a + x*c + x*d + y*z - 2*y*t - y*s + y*b + y*c - z*w + w*s + w*a - w*b - w*c + 2*t*s - t*c + u*a - u*d + v*a, 2*x^2 + x*y - x*w + x*t + x*u + 2*x*v + 2*x*r + x*s + x*a - x*c - y*t + y*a - y*d + z^2 - z*w - z*s + z*d + w*v + w*a + w*b + t*u + t*s - t*a + u*v + v^2 - v*s - v*a + v*b, x*y - x*w - x*t - x*u - x*r + x*s + x*a + x*b - x*c + x*d + y^2 - y*t + y*v - y*r + y*b + y*c - z*w + 2*z*r + z*s - w*u - w*v - w*r + 2*w*s + w*a - w*b - w*c - t*r - t*c + u*s + u*a - u*c - u*d + v*a, 2*x^2 + x*z + x*w - x*u - x*v + x*r - x*s - x*a - x*b + x*c + x*d + y*z - y*v + y*a - 2*y*b - y*d - z*a - w^2 - w*t + w*b - t*u + t*s + t*a - u*v - u*s + u*a + u*c - u*d - v^2 - v*b - v*d, x^2 + 3*x*y + x*r + x*s - x*a - x*b + x*c - y*s + y*a - y*b + y*c - y*d + z^2 + z*r + z*a + 2*z*d - w^2 + w*s + w*b - w*c + w*d + t*c - u*s + u*c + v*r - v*a - v*d, x*u - x*v - x*r - x*b - x*d - y^2 + y*t - y*u + y*r - y*c + z*w + z*a + z*c + z*d + w*u - w*s - w*a - w*d + t*r - t*a - t*d + u*r - 2*u*s + u*c + v*r - v*s, x^2 + x*y + x*z - x*w + x*u + x*v + x*r - x*s - x*d - y*z + y*s - y*b - y*c + z^2 - z*s + z*b + z*c, x*u - x*v - x*r + 2*x*a - x*b + x*d - y^2 - y*z + y*w + y*t - y*u + y*v + y*r - y*s + z^2 - z*v + 2*z*r + z*s - t*s + u*r - u*c, 3*x^2 + y*z - y*t + y*v - z*w - z*u - z*v, x*r + x*s - x*a + x*d + y*t + y*a - y*d - z*t - z*a + w*d - t*r + t*s + t*a + t*d, x*y - x*z - x*r + x*s - 2*x*a + y^2 - y*z + y*w - y*v - y*r - y*a - y*b + y*c + y*d - z*w + z*v + z*s + z*b - z*c - z*d + w*a + t*r + 2*v*r + v*s, 2*x*y - x*w - x*t - x*u - x*s - x*a - x*c + y^2 - y*w + y*t - y*v - y*r + y*s - 2*y*b - z*w - z*u - z*v - 2*z*r - z*s - z*b + w^2 - w*u - w*v - t*a - t*d, x*w - x*t + x*u + 2*x*r - x*s - 3*x*a - x*c + y*z - y*v + y*s - y*b - y*c - z*t + z*c + w*t + w*v - w*s + w*b + w*c - t*r - t*s + t*a + t*c + t*d - u*v - v^2 + v*c, 2*x^2 - x*z - 2*x*w + x*a - x*d - y*z - y*t + y*s + y*a - y*b - y*c - y*d + 2*z^2 - z*t - z*s - z*a + z*b + w*t + w*v - w*a + w*d + u*v - u*a + u*d - v*a - v*c, x*r + x*s - x*a - x*d - 2*y*a - 2*y*d - z*r - z*s + z*a + z*d + w*r + w*s + w*d + t*a + t*d - 2*u*a - v*a, x*r + x*s - x*a + y*r + 2*y*a - 2*z*r - z*s - z*a + w*r + w*a + t*r + t*s + v*a + v*d - r^2 - r*a - r*b + r*c - r*d + s^2 + s*a - s*b - s*c + a*b + a*c + b*d, x^2 + x*y + x*z - x*w + x*u + x*v + x*r - x*s - x*a - y*z + y*w - y*s - y*a - 2*y*d + z^2 - z*w + z*s - z*a + z*d - w*r + 2*r*s + r*a - r*b - r*c + 2*r*d - 3*s*a + s*b - s*c - s*d + a*b + a*c + b*d, x*z + x*w + x*u + x*v + x*r - 2*x*s + x*a - x*c + x*d + 3*y^2 - y*z + y*w + 3*y*u - y*v + y*r - 3*y*b - y*c - 2*y*d + z*u + z*v - 2*z*r - w*u + w*r - w*s + w*c - w*d + t*s - t*a - t*c - t*d + u^2 + u*v + u*r - u*s - u*d - v*s, 3*x*z - x*t + x*u + 3*x*v + x*a - x*b + x*c - y*r + y*a + y*b - y*c - 2*y*d + z^2 - 2*z*t + z*s - z*a - z*b - z*d + w*a - w*b - t*u + t*r + t*s + t*a - 2*t*c + u*s + u*a - u*c - u*d - v^2 - v*r + v*s + v*a - v*b - v*c + r^2 + r*d - s^2 - s*a - a^2 - a*d, 3*x^2 - x*y - x*w + x*u + x*v + x*r + x*s + x*b - x*c - y^2 + y*z + y*w - y*u - y*v + y*r - y*s - 2*y*a - y*b - y*d - z*w - 2*z*r - z*c - w*s + w*a + w*c + w*d - t*s + t*c - u*s - u*c + u*d - v*a - 2*v*c - 2*r^2 + 2*r*s - 2*r*c + 2*s^2 + 2*s*b + a^2 + 3*a*d - 2*b*c + d^2, 2*x^2 + x*y - x*z - x*w + 2*x*t - x*v + 2*x*r + x*a + x*b - x*c + y*z + y*w + 3*y*t - 2*y*b - z^2 + z*w - z*t + 2*z*u + z*r + z*s - z*a - w^2 - w*t + w*v - w*r + w*s + w*a + w*b - w*d + t*u + t*v + t*r - t*s - t*d - u*s + u*a + u*c - u*d - v*s - v*b - v*d, x^2 + 2*x*y - x*z - 2*x*w - x*t + x*u - 2*x*v - x*s + x*a - 2*x*b + x*c + x*d + y*z - y*t - 2*y*v + y*r + y*s + 2*y*d - 2*z^2 + z*t - 2*z*u + z*v + 2*z*r - 2*z*s + w*t + w*b + w*d - t*u - t*v - t*a + 2*t*c + v*s - v*a + v*b - v*d - r^2 - r*a - r*d + s*a + s*b + s*c + s*d + a^2 - a*b + a*d - b*d, x^2 - x*y + 2*x*z - x*w - 4*x*t + x*u + 2*x*v + 2*x*r + x*s - x*b + 2*x*d - y^2 + y*z + 2*y*w - y*t - y*s - y*a + 2*y*b - y*d + z^2 - 2*z*w + z*u - z*r + z*s + z*a - 2*z*b + z*c + z*d - w^2 - 2*w*t + 2*w*u + w*v - w*s - t*r + t*s + t*a + t*d + v^2 - v*r - v*s + v*c + r^2 + r*a + r*b - r*c + r*d - s*a - s*d, 2*x*z - 2*x*w + 2*x*t + 2*x*d + y^2 - y*z + 3*y*w - y*t - y*v + y*s + y*a + y*b - z*t - z*u + z*v - z*r - z*b + z*c + 3*w*t - w*u + w*v + w*r - 2*w*s - 2*t*u - t*v - t*r + t*s + t*a + t*d - u*s + u*a + u*c - u*d - 2*v^2 - v*r + v*a + v*c - r^2 + r*s - r*a - r*b + s*b + s*d + a^2 + a*d, x^2 + x*z - x*t + 2*x*a + 2*x*b - y^2 - 2*y*w + y*t - y*s + y*b + 3*z^2 - z*w - z*t + z*u + z*v + z*r + 2*z*s - z*a - z*b - z*c - z*d - w*t + 2*w*u - w*v + w*s + w*d + t*u + t*v - t*s + t*a + u*v + 2*u*b + v^2 - v*r - v*c - v*d + r^2 - r*s + r*b - s^2 + s*a + s*c - a^2 - a*b - a*c - a*d - b*d, 2*x^2 + x*z + 2*x*w + 2*x*t + 3*x*u - x*v - x*r + x*s + x*a + x*b + x*c + x*d - y^2 + y*z - 3*y*w + y*u + 3*y*r + y*a + y*c + z^2 + z*u - z*v - z*r + z*a - z*c + w^2 + w*t - w*u + w*r + w*b - w*d + 3*t*u + t*v - t*r + t*s - 2*t*a - t*d - u*v - u*r + u*a + u*c - u*d - v*a + v*b - v*d - 2*r^2 - r*d + 2*s^2 + s*a + 2*a^2 + 2*a*d, 2*x^2 + x*y - 2*x*z + x*t - x*u - x*v - 2*x*r - x*s - x*a - x*d + 3*y*w + y*r + 2*y*d - z^2 + 2*z*w + 2*z*t - 3*z*u + 3*z*v - z*r - z*s + z*a + z*c - w^2 - w*u + w*v - w*s - w*a - w*d + t*u - 2*t*r - t*s - t*a + u*v - 2*u*b + v^2 - v*a + v*c - r^2 + r*s - r*a - r*d - s^2 + s*a + s*b + s*c + s*d + a^2 - a*b + a*d - b*d, x^2 - x*z - 2*x*w - 2*x*u + x*v + x*r - 2*x*s - x*a + x*c + 2*y^2 - 2*y*z - y*t + 3*y*u - y*v + 2*y*r - y*a + y*b - y*c - 2*z^2 + z*w - z*v + 2*z*r - z*s - z*a + z*b + 2*w*u + w*v - w*r + w*s - w*a - w*b + w*c + t*u - t*r - t*a + t*c + u^2 - 2*u*v + u*r - u*a - u*c + v*r - v*s + v*b - r*s + r*b - r*d + s^2 + s*a - s*b, x*y + 2*x*z + x*u + x*v + x*r + x*a + 2*x*b - x*c - y^2 - 2*y*z - y*w - 2*y*t - y*v - 2*y*c + 2*y*d - z*u - z*s - z*d - 2*w^2 + 2*w*t - 2*w*u - 2*w*v + w*r - w*s + 2*w*b + 2*t*u + t*v + t*s - t*a - t*d + u*s - u*c + u*d + v^2 - v*r - v*c + v*d + r^2 - r*s + r*c - s^2 - s*b - a^2 - a*d + b*c, x^2 + x*z - x*t + x*s - x*a - x*c + x*d + 3*y^2 + y*w - y*t - 4*y*u + y*r - 2*y*b + y*c + y*d - 2*z*w + z*t + z*v - 2*z*r - z*a + z*c - z*d - w*t + w*r + t*u - 2*t*r + t*d - 3*u^2 + u*s - u*a + 2*v^2 - v*a + v*b + 2*v*c + s*a, 2*x^2 + 3*x*z + x*w + x*u - x*v + x*s + x*a + x*b + x*d + y*z - y*w + y*t - y*v + y*r + y*b + y*c + 2*z*w - z*t - z*u - z*v + z*r + 2*z*s - z*a - w^2 - w*t - w*u - w*r + w*s - 3*w*b - t*u + 3*t*v + u*v + u*r - u*a + u*b + u*c - u*d + v*r + v*a + v*b + v*c + r^2 + r*d - s^2 - s*a - a^2 - a*b - a*d, 2*x^2 + 2*x*y + x*z + 2*x*w + x*t - x*v - x*r - 2*x*s + x*d + 3*y*w + y*v + y*r + y*a + y*c + y*d - 2*z^2 + z*w - 2*z*t - z*u + z*v - z*r + z*a - z*c + w*u + w*v - w*r + w*s - w*a - w*b - t*u - 4*t*v - t*r - t*s - u*v + u*s - u*b - u*c + v^2 + v*a - v*b - r^2 + r*b - 2*r*d + s^2 + 2*s*a - s*b + a^2 + a*d - b*d, x*y + x*z + x*w + x*t + 3*x*u + x*r + x*s - x*a + x*d - 2*y*z - 2*y*w + y*t + y*u + y*v - y*r + 2*y*b + y*d + 2*z*t + z*r + z*a + 2*w^2 + w*t + w*v - w*r + w*b - w*d - 3*t*u + 2*t*v - t*r + 2*t*s + u*v - u*r + u*s + u*a - u*b + v^2 - v*s - v*a + r^2 - r*s + r*a + r*b - s*b - a^2 - a*b - a*c, x^2 + x*y - 2*x*z + 2*x*w + x*t + x*u - 2*x*v - x*r - 2*x*s - 3*x*a + x*b - x*c - x*d + y*z - 2*y*w - 2*y*t + y*u - y*s - y*a - 2*y*b + z^2 + z*w + z*t - z*u - z*r + z*b + z*d - w^2 + w*u - w*v - w*a - w*d - 7*t^2 - t*u - t*v - t*r + t*a - 2*t*b + 2*u*v - u*r - u*s + 2*u*c - v^2 + v*b + v*c + r*c + r*d - s^2 - 2*s*a + s*b - s*d + 2*a*c - a*d + 2*b^2 + b*c, 2*x^2 + x*y + 2*x*z + 4*x*w - x*t - 2*x*u + 2*x*v + 4*x*r + x*s - 2*x*c - y^2 - y*z + y*t + y*u - y*v + y*r - y*s - y*a + 2*y*b + y*d - z*t + 2*z*u - z*v + z*s + z*b + z*c + w*t + w*a + w*c - w*d + 3*t*u + t*v + t*r + t*c + t*d + 2*u*v - u*r + u*s - 2*v^2 - v*a + v*b - r*a - r*b + r*d + 3*s*b + s*d - a^2 - a*c - 2*b*c + d^2, x*w + x*u - x*v + x*r - x*s - x*a + x*b + 3*x*c + y^2 - y*z - 2*y*t - 4*y*u + y*r + y*s - y*c + 2*y*d + 3*z^2 - z*u - 2*z*r + z*s + z*c - z*d - 2*w^2 - w*t + w*u + w*s - w*a - w*b - w*c + w*d + 3*t*u + t*v - t*r + t*s - 2*t*a + t*c - t*d + 6*u^2 - 2*u*r + u*s + u*c + u*d + v^2 + v*b + v*c - v*d - r^2 - r*s + r*a + r*b - r*d - 3*s*b - s*d + 3*a^2 + a*c + a*d + 2*b*c - d^2, 2*x^2 - 2*x*z - x*w + 3*x*t - 2*x*v - x*r - 2*x*s + x*a + x*c - x*d + y^2 + 4*y*z - 2*y*w + 2*y*t - y*v - 2*y*r + y*s + 2*y*b - y*c + 3*z^2 - z*w + 3*z*t - z*u + z*r - 2*z*s - 6*w^2 + w*t - 2*w*s - w*b + w*c + t*u + t*s - t*a - u*v - v*r + v*a + v*b + v*d + r^2 - s^2 - s*a - s*d - a^2 + a*c - a*d + c*d, 2*x^2 - x*y - x*z + x*w - 2*x*t + x*u + x*r - x*s + x*c + 7*y^2 + 4*y*z + 5*y*w + 3*y*t + y*u - 6*y*v + y*s + y*a + 2*y*b - y*c - y*d + 4*z^2 + 6*z*w + z*t + 2*z*u - 6*z*v + z*r + 4*z*b - z*c - z*d + 5*w^2 + w*t - 2*w*u - 5*w*v - w*s + 4*w*b - 3*w*d + 2*t*u - 2*t*v + 3*t*a - t*c + 2*t*d - 2*u^2 + u*v - 2*u*r + u*s + u*a - 3*u*b + 6*v^2 - v*s - 2*v*a - 3*v*b - v*c - r^2 - r*s + r*a + r*b + r*c - r*d + s^2 + 2*s*c - s*d - a^2 + 2*a*b + a*c + 3*a*d - b*c + b*d - 2*c^2 + c*d + 2*d^2]

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

Map of degree 4 from the canonical model of this modular curve to the canonical model of the modular curve 30.48.3.e.2 :

$\displaystyle X$	$=$	$\displaystyle 5x$
$\displaystyle Y$	$=$	$\displaystyle -5x+3a+2d$
$\displaystyle Z$	$=$	$\displaystyle -5x+a-d$

Equation of the image curve:

$0$

$=$

$ 36X^{4}-2X^{3}Y+3X^{2}Y^{2}+2XY^{3}-10X^{3}Z+12X^{2}YZ+12XY^{2}Z+2Y^{3}Z-9X^{2}Z^{2}+3Y^{2}Z^{2}-2XZ^{3}-2YZ^{3} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
120.96.3-30.e.2.5	$120$	$4$	$4$	$3$	$?$
120.192.7-60.g.1.14	$120$	$2$	$2$	$7$	$?$
120.192.7-60.g.1.26	$120$	$2$	$2$	$7$	$?$

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