LMFDB - Modular curve 50.300.14.g.1

Invariants

Level:	$50$	$\SL_2$-level:	$50$	Newform level:	$500$
Index:	$300$	$\PSL_2$-index:	$300$
Genus:	$14 = 1 + \frac{ 300 }{12} - \frac{ 20 }{4} - \frac{ 0 }{3} - \frac{ 14 }{2}$
Cusps:	$14$ (of which $5$ are rational)	Cusp widths	$10^{10}\cdot50^{4}$	Cusp orbits	$1^{5}\cdot4\cdot5$
Elliptic points:	$20$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$4 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 8$
Rational cusps:	$5$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	50A14
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	50.300.14.2

Level structure

$\GL_2(\Z/50\Z)$-generators:	$\begin{bmatrix}7&39\\10&3\end{bmatrix}$, $\begin{bmatrix}9&21\\25&18\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 50-isogeny field degree:	$15$
Cyclic 50-torsion field degree:	$60$
Full 50-torsion field degree:	$6000$

Jacobian

Conductor:	$2^{20}\cdot5^{36}$
Simple:	no
Squarefree:	yes
Decomposition:	$2\cdot4\cdot8$
Newforms:	25.2.d.a, 100.2.c.a, 500.2.i.a

Models

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

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

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

Rational points

This modular curve has 5 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:1/2:1/2:0:1:0:0:0:0:1:0)$, $(0:0:0:0:-1/2:1/2:1:1/2:0:0:0:0:1:0)$, $(0:0:0:0:1:-2:2:0:0:0:0:0:1:0)$, $(0:0:0:0:-1:-2:-1:-1:0:0:0:0:1:0)$, $(0:0:0:0:-1/2:0:1/2:1:0:0:0:0:0:0)$

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 25.150.4.b.1 :

$\displaystyle X$	$=$	$\displaystyle -y$
$\displaystyle Y$	$=$	$\displaystyle -z$
$\displaystyle Z$	$=$	$\displaystyle -x-y+z+w$
$\displaystyle W$	$=$	$\displaystyle x$

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

Cover information

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

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
10.60.2.c.1	$10$	$5$	$5$	$2$	$0$	$4\cdot8$
25.150.4.b.1	$25$	$2$	$2$	$4$	$0$	$2\cdot8$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
50.600.37.c.1	$50$	$2$	$2$	$37$	$0$	$1^{3}\cdot8\cdot12$
50.600.37.r.1	$50$	$2$	$2$	$37$	$0$	$1^{3}\cdot8\cdot12$
50.900.50.g.1	$50$	$3$	$3$	$50$	$0$	$1^{2}\cdot2\cdot4^{2}\cdot8^{3}$
50.1500.96.d.1	$50$	$5$	$5$	$96$	$2$	$2^{3}\cdot4^{5}\cdot8^{2}\cdot16\cdot24$
50.1500.96.d.2	$50$	$5$	$5$	$96$	$2$	$2^{3}\cdot4^{5}\cdot8^{2}\cdot16\cdot24$

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