LMFDB - Modular curve 24.256.13.d.1

Invariants

Level:	$24$	$\SL_2$-level:	$24$	Newform level:	$192$
Index:	$256$	$\PSL_2$-index:	$256$
Genus:	$13 = 1 + \frac{ 256 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)	Cusp widths	$8^{8}\cdot24^{8}$	Cusp orbits	$2^{2}\cdot4^{3}$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$4 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 8$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	24G13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.256.13.4

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}1&19\\12&7\end{bmatrix}$, $\begin{bmatrix}11&13\\9&10\end{bmatrix}$, $\begin{bmatrix}13&8\\0&5\end{bmatrix}$, $\begin{bmatrix}14&11\\15&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup:	$D_6^2:C_2$
Contains $-I$:	yes
Quadratic refinements:	24.512.13-24.d.1.1, 24.512.13-24.d.1.2, 24.512.13-24.d.1.3, 24.512.13-24.d.1.4, 24.512.13-24.d.1.5, 24.512.13-24.d.1.6, 24.512.13-24.d.1.7, 24.512.13-24.d.1.8, 120.512.13-24.d.1.1, 120.512.13-24.d.1.2, 120.512.13-24.d.1.3, 120.512.13-24.d.1.4, 120.512.13-24.d.1.5, 120.512.13-24.d.1.6, 120.512.13-24.d.1.7, 120.512.13-24.d.1.8, 168.512.13-24.d.1.1, 168.512.13-24.d.1.2, 168.512.13-24.d.1.3, 168.512.13-24.d.1.4, 168.512.13-24.d.1.5, 168.512.13-24.d.1.6, 168.512.13-24.d.1.7, 168.512.13-24.d.1.8, 264.512.13-24.d.1.1, 264.512.13-24.d.1.2, 264.512.13-24.d.1.3, 264.512.13-24.d.1.4, 264.512.13-24.d.1.5, 264.512.13-24.d.1.6, 264.512.13-24.d.1.7, 264.512.13-24.d.1.8, 312.512.13-24.d.1.1, 312.512.13-24.d.1.2, 312.512.13-24.d.1.3, 312.512.13-24.d.1.4, 312.512.13-24.d.1.5, 312.512.13-24.d.1.6, 312.512.13-24.d.1.7, 312.512.13-24.d.1.8
Cyclic 24-isogeny field degree:	$6$
Cyclic 24-torsion field degree:	$24$
Full 24-torsion field degree:	$288$

Jacobian

Conductor:	$2^{76}\cdot3^{11}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{7}\cdot2\cdot4$
Newforms:	48.2.a.a, 64.2.a.a$^{2}$, 192.2.a.a, 192.2.a.c, 192.2.a.d$^{2}$, 192.2.c.a, 192.2.c.b

Models

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

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

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

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 24.64.4.a.1 :

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

Equation of the image curve:

$0$	$=$	$ 2X^{2}-8XY-Y^{2}+12XZ-2YZ+Z^{2}+8XW+10YW+2ZW-W^{2} $
	$=$	$ X^{3}-2X^{2}Y-XY^{2}+3X^{2}Z+XZ^{2}+2X^{2}W+4XYW+Y^{2}W+YZW-XW^{2}-YW^{2} $

Modular covers

Cover information

Click on a modular curve in the diagram to see information about it.

See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
12.64.1.b.1	$12$	$4$	$4$	$1$	$0$	$1^{6}\cdot2\cdot4$
24.128.7.c.1	$24$	$2$	$2$	$7$	$1$	$2\cdot4$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
24.768.41.jg.2	$24$	$3$	$3$	$41$	$3$	$1^{14}\cdot2^{3}\cdot4^{2}$
24.768.49.c.1	$24$	$3$	$3$	$49$	$5$	$1^{18}\cdot2^{5}\cdot4^{2}$
48.1024.69.d.1	$48$	$4$	$4$	$69$	$15$	$1^{16}\cdot2^{12}\cdot4^{4}$

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