LMFDB - Modular curve 60.480.13-60.nm.1.4

Invariants

Level:	$60$	$\SL_2$-level:	$60$	Newform level:	$1200$
Index:	$480$	$\PSL_2$-index:	$240$
Genus:	$13 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)	Cusp widths	$5^{4}\cdot10^{4}\cdot15^{4}\cdot30^{4}$	Cusp orbits	$4^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$4 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 6$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	30N13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.480.13.1157

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}1&27\\36&29\end{bmatrix}$, $\begin{bmatrix}5&34\\24&37\end{bmatrix}$, $\begin{bmatrix}23&9\\54&25\end{bmatrix}$, $\begin{bmatrix}29&24\\24&11\end{bmatrix}$, $\begin{bmatrix}41&32\\42&7\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.240.13.nm.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree:	$12$
Cyclic 60-torsion field degree:	$192$
Full 60-torsion field degree:	$4608$

Jacobian

Conductor:	$2^{27}\cdot3^{9}\cdot5^{26}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{13}$
Newforms:	50.2.a.b$^{2}$, 75.2.a.a$^{2}$, 75.2.a.b$^{2}$, 150.2.a.b, 400.2.a.f$^{2}$, 1200.2.a.c$^{2}$, 1200.2.a.g, 1200.2.a.m

Models

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

$ 0 $	$=$	$ x t + x c - y t + y u - y s - 2 z t + z u - z s + u v - v s + r s + r c $
	$=$	$x t + x u - x s - x c + x d + y t + y s + y b + y c - z t + z s - u v + v s - r s - r c - s a$
	$=$	$2 x t - x u + x s + x c - y u + y s + y c + z u - u v$
	$=$	$x t - x u + x s + x b + y d - z t - z u - u v - v s - v b - v d - r s - r b + a c$
	$=$	$\cdots$

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

Rational points

This modular curve has no $\Q_p$ points for $p=13,29$, 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 30.120.7.h.1 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle z$
$\displaystyle W$	$=$	$\displaystyle v$
$\displaystyle T$	$=$	$\displaystyle -w$
$\displaystyle U$	$=$	$\displaystyle -r$
$\displaystyle V$	$=$	$\displaystyle -a$

Equation of the image curve:

$0$	$=$	$ XY-2Y^{2}+YZ+Z^{2}-XW+YW-YT+YV-WV $
	$=$	$ 2X^{2}+XY-2XZ-2Z^{2}+2XW-YW+ZW+W^{2}+XT-YT+WT-XU-YU+ZU+WU-TU-U^{2}-ZV $
	$=$	$ X^{2}-3XZ+YZ+Z^{2}+2XW-2ZW-XT-WT-T^{2}-XU+YU-3ZU-2TU+XV+YV-WV-UV $
	$=$	$ X^{2}-XY-3YZ+YW-2ZW-YT-ZT-WT-T^{2}-XU-3ZU-2TU+XV-YV+TV-UV $
	$=$	$ 2XZ+YZ+2XW-2YW+ZW+2XT-2YT+ZT+WT+T^{2}+YU+2ZU+TU+YV+ZV-WV $
	$=$	$ 2XY+2XZ+Z^{2}+2XW-ZW+XT+2ZT-WT+YU-2ZU-TU-WV $
	$=$	$ X^{2}-XZ+YZ+XW-2YW+ZW+XT-2ZT-WT-XU+2YU+TU+XV-YV+2ZV+WV-TV-UV $
	$=$	$ 2XY-3XZ-2YZ-2Z^{2}-XW-ZW-2XT-2YT-2ZT-T^{2}+YU-ZU-TU-ZV-WV $
	$=$	$ 2X^{2}-2XZ+YZ-Z^{2}-XW-XT+2ZT+T^{2}-2XU+2YU-2XV+YV-3ZV-WV-2TV+2UV $
	$=$	$ X^{2}-XZ+YZ-2YW-W^{2}+3XT-2YT-ZT+T^{2}+3XU+YU+ZU-WU+2TU+U^{2}-4XV+YV-WV-TV-UV-V^{2} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.240.7-30.h.1.12	$60$	$2$	$2$	$7$	$0$	$1^{6}$
60.240.7-30.h.1.14	$60$	$2$	$2$	$7$	$0$	$1^{6}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.960.29-60.bi.1.20	$60$	$2$	$2$	$29$	$4$	$1^{16}$
60.960.29-60.ef.1.16	$60$	$2$	$2$	$29$	$4$	$1^{16}$
60.960.29-60.fp.1.1	$60$	$2$	$2$	$29$	$4$	$1^{16}$
60.960.29-60.fr.1.6	$60$	$2$	$2$	$29$	$4$	$1^{16}$
60.960.29-60.iz.1.9	$60$	$2$	$2$	$29$	$8$	$1^{16}$
60.960.29-60.jb.1.11	$60$	$2$	$2$	$29$	$11$	$1^{16}$
60.960.29-60.jd.1.5	$60$	$2$	$2$	$29$	$11$	$1^{16}$
60.960.29-60.jf.1.2	$60$	$2$	$2$	$29$	$8$	$1^{16}$
60.960.29-60.oe.1.12	$60$	$2$	$2$	$29$	$6$	$1^{16}$
60.960.29-60.og.1.16	$60$	$2$	$2$	$29$	$5$	$1^{16}$
60.960.29-60.oh.1.8	$60$	$2$	$2$	$29$	$5$	$1^{16}$
60.960.29-60.ok.1.3	$60$	$2$	$2$	$29$	$6$	$1^{16}$
60.960.29-60.or.1.16	$60$	$2$	$2$	$29$	$2$	$1^{16}$
60.960.29-60.ot.1.10	$60$	$2$	$2$	$29$	$4$	$1^{16}$
60.960.29-60.ou.1.4	$60$	$2$	$2$	$29$	$4$	$1^{16}$
60.960.29-60.ox.1.7	$60$	$2$	$2$	$29$	$2$	$1^{16}$
60.960.33-60.ph.1.10	$60$	$2$	$2$	$33$	$15$	$1^{20}$
60.960.33-60.pk.1.13	$60$	$2$	$2$	$33$	$10$	$1^{20}$
60.960.33-60.pp.1.14	$60$	$2$	$2$	$33$	$3$	$1^{20}$
60.960.33-60.ps.1.9	$60$	$2$	$2$	$33$	$6$	$1^{20}$
60.960.33-60.qs.1.15	$60$	$2$	$2$	$33$	$10$	$1^{20}$
60.960.33-60.qu.1.12	$60$	$2$	$2$	$33$	$5$	$1^{20}$
60.960.33-60.ra.1.11	$60$	$2$	$2$	$33$	$6$	$1^{20}$
60.960.33-60.rc.1.24	$60$	$2$	$2$	$33$	$9$	$1^{20}$
60.1440.37-60.nm.1.27	$60$	$3$	$3$	$37$	$5$	$1^{24}$
60.1440.45-60.d.1.9	$60$	$3$	$3$	$45$	$12$	$1^{28}\cdot2^{2}$

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