LMFDB - Modular curve 25.600.12-25.c.1.2

Invariants

Level:	$25$	$\SL_2$-level:	$25$	Newform level:	$125$
Index:	$600$	$\PSL_2$-index:	$300$
Genus:	$12 = 1 + \frac{ 300 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 28 }{2}$
Cusps:	$28$ (none of which are rational)	Cusp widths	$5^{20}\cdot25^{8}$	Cusp orbits	$2^{5}\cdot4^{2}\cdot5^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$4 \le \gamma \le 5$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	25A12
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	25.600.12.2

Level structure

$\GL_2(\Z/25\Z)$-generators:	$\begin{bmatrix}7&20\\15&6\end{bmatrix}$, $\begin{bmatrix}24&20\\5&1\end{bmatrix}$
$\GL_2(\Z/25\Z)$-subgroup:	$C_5^2:C_{20}$
Contains $-I$:	no $\quad$ (see 25.300.12.c.1 for the level structure with $-I$)
Cyclic 25-isogeny field degree:	$5$
Cyclic 25-torsion field degree:	$20$
Full 25-torsion field degree:	$500$

Jacobian

Conductor:	$5^{32}$
Simple:	no
Squarefree:	yes
Decomposition:	$4\cdot8$
Newforms:	25.2.d.a, 125.2.e.b

Models

Canonical model in $\mathbb{P}^{ 11 }$ defined by 45 equations

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

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

Rational points

This modular curve has no $\Q_p$ points for $p=2,3,13,17$, and therefore no rational points.

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 25.150.4.b.1 :

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

Equation of the image curve:

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

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{arith}}(5)$	$5$	$5$	$5$	$0$	$0$	full Jacobian

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
25.3000.76-25.c.1.2	$25$	$5$	$5$	$76$	$2$	$2^{2}\cdot4^{5}\cdot8^{3}\cdot16$
25.3000.76-25.c.2.2	$25$	$5$	$5$	$76$	$2$	$2^{2}\cdot4^{5}\cdot8^{3}\cdot16$
50.1200.37-50.c.1.2	$50$	$2$	$2$	$37$	$0$	$1^{3}\cdot2\cdot8\cdot12$
50.1200.37-50.p.1.2	$50$	$2$	$2$	$37$	$0$	$1^{3}\cdot2\cdot8\cdot12$
50.1800.48-50.c.1.1	$50$	$3$	$3$	$48$	$0$	$1^{2}\cdot2\cdot4^{2}\cdot8^{3}$

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