LMFDB - Modular curve 210.432.11-30.a.1.1

Invariants

Level:	$210$	$\SL_2$-level:	$30$	Newform level:	$90$
Index:	$432$	$\PSL_2$-index:	$216$
Genus:	$11 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (of which $8$ are rational)	Cusp widths	$3^{4}\cdot6^{4}\cdot15^{4}\cdot30^{4}$	Cusp orbits	$1^{8}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 11$
Rational cusps:	$8$
Rational CM points:	none

Level structure

$\GL_2(\Z/210\Z)$-generators:	$\begin{bmatrix}3&20\\8&117\end{bmatrix}$, $\begin{bmatrix}47&90\\87&209\end{bmatrix}$, $\begin{bmatrix}49&150\\144&67\end{bmatrix}$, $\begin{bmatrix}79&90\\105&193\end{bmatrix}$, $\begin{bmatrix}103&180\\63&103\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 30.216.11.a.1 for the level structure with $-I$)
Cyclic 210-isogeny field degree:	$8$
Cyclic 210-torsion field degree:	$384$
Full 210-torsion field degree:	$645120$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

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

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

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(1/3:1/3:-1/3:-1/3:0:0:1/3:-1/3:0:1/3:1)$, $(0:-1:0:0:0:-1:1:0:0:1:0)$, $(1:1:0:0:0:-1:0:0:1:0:0)$, $(1/2:0:0:-1/2:-1/2:0:0:0:-1/2:-1/2:1)$, $(0:0:0:1:0:1:0:1:0:0:0)$, $(0:0:1:0:1:1:0:1:-1:1:1)$, $(0:0:0:0:0:0:0:1:0:0:0)$, $(0:0:0:0:0:0:0:0:1:0:0)$

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 15.72.3.a.1 :

$\displaystyle X$	$=$	$\displaystyle y-z-w+u$
$\displaystyle Y$	$=$	$\displaystyle x+2z-w+u$
$\displaystyle Z$	$=$	$\displaystyle x-y$

Equation of the image curve:

$0$

$=$

$ X^{2}Y^{2}+X^{3}Z-Y^{3}Z-XYZ^{2}+5Z^{4} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(2)$	$2$	$144$	$72$	$0$	$0$
$X_0(5)$	$5$	$72$	$36$	$0$	$0$
21.24.0-3.a.1.1	$21$	$18$	$18$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
42.72.0-6.a.1.1	$42$	$6$	$6$	$0$	$0$
105.144.3-15.a.1.1	$105$	$3$	$3$	$3$	$?$
210.144.3-30.a.1.13	$210$	$3$	$3$	$3$	$?$
210.144.3-30.a.1.14	$210$	$3$	$3$	$3$	$?$

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