LMFDB - Modular curve 210.360.12-30.a.1.1

Invariants

Level:	$210$	$\SL_2$-level:	$30$	Newform level:	$450$
Index:	$360$	$\PSL_2$-index:	$180$
Genus:	$12 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)	Cusp widths	$15^{4}\cdot30^{4}$	Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 12$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 12$
Rational cusps:	$4$
Rational CM points:	none

Level structure

$\GL_2(\Z/210\Z)$-generators:	$\begin{bmatrix}13&15\\18&139\end{bmatrix}$, $\begin{bmatrix}55&141\\66&73\end{bmatrix}$, $\begin{bmatrix}83&149\\180&169\end{bmatrix}$, $\begin{bmatrix}199&22\\72&155\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 30.180.12.a.1 for the level structure with $-I$)
Cyclic 210-isogeny field degree:	$48$
Cyclic 210-torsion field degree:	$2304$
Full 210-torsion field degree:	$774144$

Models

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

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

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

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

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 30.45.3.a.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle v$
$\displaystyle Z$	$=$	$\displaystyle -2y+2z-2t+v-2r+s-a+b-4c$

Equation of the image curve:

$0$

$=$

$ 7X^{4}-6X^{3}Y+5X^{2}Y^{2}-5XY^{3}-Y^{4}-3X^{3}Z-3X^{2}YZ-5XY^{2}Z+2Y^{3}Z-3X^{2}Z^{2}+3XYZ^{2}-Y^{2}Z^{2}-XZ^{3} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(2)$	$2$	$120$	$60$	$0$	$0$
$X_{S_4}(5)$	$5$	$72$	$36$	$0$	$0$
21.24.0-3.a.1.1	$21$	$15$	$15$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
42.72.0-6.a.1.1	$42$	$5$	$5$	$0$	$0$
105.120.4-15.a.1.1	$105$	$3$	$3$	$4$	$?$
210.120.4-30.b.1.2	$210$	$3$	$3$	$4$	$?$
210.120.4-30.b.1.4	$210$	$3$	$3$	$4$	$?$

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