LMFDB - Modular curve 30.216.13.n.2

Invariants

Level:	$30$	$\SL_2$-level:	$30$	Newform level:	$180$
Index:	$216$	$\PSL_2$-index:	$216$
Genus:	$13 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (none of which are rational)	Cusp widths	$6^{6}\cdot30^{6}$	Cusp orbits	$2^{4}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$6$
$\overline{\Q}$-gonality:	$6$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	30H13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	30.216.13.25

Level structure

$\GL_2(\Z/30\Z)$-generators:	$\begin{bmatrix}3&25\\22&3\end{bmatrix}$, $\begin{bmatrix}23&5\\16&29\end{bmatrix}$, $\begin{bmatrix}27&5\\2&9\end{bmatrix}$
$\GL_2(\Z/30\Z)$-subgroup:	$F_5\times Q_8:C_4$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 30-isogeny field degree:	$4$
Cyclic 30-torsion field degree:	$32$
Full 30-torsion field degree:	$640$

Jacobian

Conductor:	$2^{10}\cdot3^{26}\cdot5^{11}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{7}\cdot2^{3}$
Newforms:	36.2.a.a$^{2}$, 45.2.b.a$^{3}$, 90.2.a.a$^{2}$, 90.2.a.b$^{2}$, 180.2.a.a

Models

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

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

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

Rational points

This modular curve has no real points, and therefore no rational points.

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 30.72.5.o.1 :

$\displaystyle X$	$=$	$\displaystyle -w$
$\displaystyle Y$	$=$	$\displaystyle -x+3y-z$
$\displaystyle Z$	$=$	$\displaystyle x-r+s+d$
$\displaystyle W$	$=$	$\displaystyle -t+3v$
$\displaystyle T$	$=$	$\displaystyle t+a+b-2c$

Equation of the image curve:

$0$	$=$	$ XY+W^{2}-WT+T^{2} $
	$=$	$ 4Y^{2}-YZ+Z^{2}+5XW $
	$=$	$ 10X^{2}-2YW-ZW-YT+2ZT $

Modular covers

Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
30.72.1.g.2	$30$	$3$	$3$	$1$	$0$	$1^{6}\cdot2^{3}$
30.72.5.o.1	$30$	$3$	$3$	$5$	$0$	$1^{4}\cdot2^{2}$
30.72.5.p.1	$30$	$3$	$3$	$5$	$0$	$1^{4}\cdot2^{2}$
30.108.4.d.2	$30$	$2$	$2$	$4$	$0$	$1^{5}\cdot2^{2}$
30.108.6.b.2	$30$	$2$	$2$	$6$	$0$	$1^{5}\cdot2$
30.108.7.b.1	$30$	$2$	$2$	$7$	$0$	$2^{3}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
30.432.25.m.2	$30$	$2$	$2$	$25$	$0$	$1^{6}\cdot2^{3}$
30.432.25.p.2	$30$	$2$	$2$	$25$	$0$	$1^{6}\cdot2^{3}$
30.432.25.y.1	$30$	$2$	$2$	$25$	$1$	$1^{6}\cdot2^{3}$
30.432.25.z.1	$30$	$2$	$2$	$25$	$0$	$1^{6}\cdot2^{3}$
30.1080.73.bu.1	$30$	$5$	$5$	$73$	$7$	$1^{24}\cdot2^{18}$
60.432.25.gc.2	$60$	$2$	$2$	$25$	$2$	$1^{6}\cdot2^{3}$
60.432.25.hn.2	$60$	$2$	$2$	$25$	$4$	$1^{6}\cdot2^{3}$
60.432.25.kv.2	$60$	$2$	$2$	$25$	$4$	$1^{6}\cdot2^{3}$
60.432.25.lk.2	$60$	$2$	$2$	$25$	$1$	$1^{6}\cdot2^{3}$
60.432.29.th.2	$60$	$2$	$2$	$29$	$4$	$1^{8}\cdot2^{4}$
60.432.29.tl.2	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.bkv.2	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.bkx.2	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.bpd.2	$60$	$2$	$2$	$29$	$4$	$1^{8}\cdot2^{4}$
60.432.29.bpf.2	$60$	$2$	$2$	$29$	$2$	$1^{8}\cdot2^{4}$
60.432.29.btd.2	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.btf.2	$60$	$2$	$2$	$29$	$3$	$1^{8}\cdot2^{4}$
60.432.29.cez.2	$60$	$2$	$2$	$29$	$2$	$1^{8}\cdot2^{4}$
60.432.29.cfb.2	$60$	$2$	$2$	$29$	$6$	$1^{8}\cdot2^{4}$
60.432.29.chd.2	$60$	$2$	$2$	$29$	$3$	$1^{8}\cdot2^{4}$
60.432.29.chf.2	$60$	$2$	$2$	$29$	$6$	$1^{8}\cdot2^{4}$
60.432.29.cib.2	$60$	$2$	$2$	$29$	$0$	$1^{8}\cdot2^{4}$
60.432.29.cid.2	$60$	$2$	$2$	$29$	$2$	$1^{8}\cdot2^{4}$
60.432.29.cjx.2	$60$	$2$	$2$	$29$	$0$	$1^{8}\cdot2^{4}$
60.432.29.ckb.2	$60$	$2$	$2$	$29$	$4$	$1^{8}\cdot2^{4}$

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