LMFDB - Modular curve $X_{\mathrm{ns}}(22)$

Invariants

Level:	$22$	$\SL_2$-level:	$22$	Newform level:	$484$
Index:	$220$	$\PSL_2$-index:	$220$
Genus:	$13 = 1 + \frac{ 220 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 10 }{2}$
Cusps:	$10$ (none of which are rational)	Cusp widths	$22^{10}$	Cusp orbits	$10$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
Analytic rank:	$4$
$\Q$-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:	22A13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	22.220.13.4

Level structure

$\GL_2(\Z/22\Z)$-generators:	$\begin{bmatrix}4&9\\13&17\end{bmatrix}$, $\begin{bmatrix}21&10\\12&11\end{bmatrix}$
$\GL_2(\Z/22\Z)$-subgroup:	$C_3\times C_{120}$
Contains $-I$:	yes
Quadratic refinements:	44.440.13-22.a.1.1, 44.440.13-22.a.1.2, 88.440.13-22.a.1.1, 88.440.13-22.a.1.2, 132.440.13-22.a.1.1, 132.440.13-22.a.1.2, 220.440.13-22.a.1.1, 220.440.13-22.a.1.2, 264.440.13-22.a.1.1, 264.440.13-22.a.1.2, 308.440.13-22.a.1.1, 308.440.13-22.a.1.2
Cyclic 22-isogeny field degree:	$36$
Cyclic 22-torsion field degree:	$360$
Full 22-torsion field degree:	$360$

Jacobian

Conductor:	$2^{18}\cdot11^{26}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{5}\cdot2^{4}$
Newforms:	121.2.a.a, 121.2.a.b, 121.2.a.c, 121.2.a.d, 484.2.a.a, 484.2.a.b, 484.2.a.c, 484.2.a.d, 484.2.a.e

Models

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

$ 0 $	$=$	$ x d + y d + z d - v b - v c - r d $
	$=$	$x d - z d + t d - v a + v b + v c + r d$
	$=$	$t a + u a - v a + v b + v c + r a$
	$=$	$x d - y d - z d - w d + v b$
	$=$	$\cdots$

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

Rational points

This modular curve has no real points, 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 $X_{\mathrm{ns}}(11)$ :

$\displaystyle X$	$=$	$\displaystyle -a$
$\displaystyle Y$	$=$	$\displaystyle b$
$\displaystyle Z$	$=$	$\displaystyle a-b-c-d$
$\displaystyle W$	$=$	$\displaystyle d$

Equation of the image curve:

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

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

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

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{ns}}(2)$	$2$	$110$	$110$	$0$	$0$	full Jacobian
$X_{\mathrm{ns}}(11)$	$11$	$2$	$2$	$4$	$1$	$1\cdot2^{4}$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{ns}}(11)$	$11$	$2$	$2$	$4$	$1$	$1\cdot2^{4}$
$X_{\mathrm{ns}}^+(22)$	$22$	$2$	$2$	$4$	$4$	$1^{3}\cdot2^{3}$
22.110.7.a.1	$22$	$2$	$2$	$7$	$1$	$1^{4}\cdot2$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
22.660.39.a.1	$22$	$3$	$3$	$39$	$12$	$1^{10}\cdot2^{8}$
22.660.41.a.1	$22$	$3$	$3$	$41$	$12$	$1^{12}\cdot2^{8}$
22.660.41.c.1	$22$	$3$	$3$	$41$	$12$	$1^{12}\cdot2^{8}$
$X_{\mathrm{ns}}(44)$	$44$	$4$	$4$	$63$	$26$	$1^{12}\cdot2^{15}\cdot4^{2}$
66.660.51.a.1	$66$	$3$	$3$	$51$	$17$	$1^{8}\cdot2^{5}\cdot4^{5}$
66.880.63.a.1	$66$	$4$	$4$	$63$	$21$	$1^{16}\cdot2^{15}\cdot4$

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