LMFDB - Modular curve 15.360.10-15.b.1.5

Invariants

Level:	$15$	$\SL_2$-level:	$15$	Newform level:	$225$
Index:	$360$	$\PSL_2$-index:	$180$
Genus:	$10 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $2$ are rational)	Cusp widths	$15^{12}$	Cusp orbits	$1^{2}\cdot2\cdot4^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$3$
$\overline{\Q}$-gonality:	$3$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	15A10
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	15.360.10.1

Level structure

$\GL_2(\Z/15\Z)$-generators:	$\begin{bmatrix}2&4\\5&11\end{bmatrix}$, $\begin{bmatrix}7&9\\0&1\end{bmatrix}$, $\begin{bmatrix}11&5\\5&1\end{bmatrix}$
$\GL_2(\Z/15\Z)$-subgroup:	$C_4\times \SD_{16}$
Contains $-I$:	no $\quad$ (see 15.180.10.b.1 for the level structure with $-I$)
Cyclic 15-isogeny field degree:	$4$
Cyclic 15-torsion field degree:	$8$
Full 15-torsion field degree:	$64$

Jacobian

Conductor:	$3^{20}\cdot5^{16}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{2}\cdot2^{4}$
Newforms:	45.2.b.a$^{2}$, 225.2.a.c, 225.2.a.d, 225.2.a.f, 225.2.b.c

Models

Canonical model in $\mathbb{P}^{ 9 }$ defined by 35 equations

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

[w*r + u^2, x*s + y*z + y*w - y*u - y*r + v*a, x*y + y*z + y*t - y*u - y*r - u*a, t*u + t*r - u^2 - v*r, y*z + y*u - y*r - z*a - w*a, y*z + y*r - z*a - w*a + u*a + r*s, x*y + y*z + y*t + y*u - u*s - r*a, y*z - y*w - y*u - y*r - z*a - u*s, x*y - y*z - 2*y*t + y*u + y*v + u*s, x*y - y*z + y*t - y*u + t*s - t*a + v*a, x*u - x*r + z*u + z*r + t*r - u^2, 2*y*w - y*u - w*s + w*a + u*a, x*y + x*s + x*a + y*t + y*u - z*a - w*s + v*s, x*y - x*a - y*t - y*v - y*r + w*s + t*a - u*s + v*a - r*a, x*w + x*r + z*w - t*u - u^2 - u*v - u*r, x*w + x*u + z*w - z*u - w*u - t*u, w*t - w*u - t*u + u*v, x*u + z*u - z*r + w*u - v*r + r^2, x^2 + x*w - x*t - x*v - z^2 - z*w + z*u - z*v + z*r + u*v + v*r, x*z + x*w + x*t - x*u - x*v - z^2 - z*w + t*v - t*r + u^2 + u*v, 2*x*w + x*u + z*u + w*u + w*v - u^2 + u*v - u*r, x*w + z*w + z*u + w^2 + u*v - u*r, x*y - y*t - y*u - y*v + y*r + z*s + w*s, x*z - x*r + z^2 + z*t + w*t + u*r + v*r, x*z + x*w - x*t - z^2 - z*w - z*u + z*v + t^2 + t*v - t*r + u^2, x^2 - x*z - x*u - x*v + x*r - z*t - w*u + w*v + t*r - v^2, 2*x*z - x*u - z*t + z*u + z*v - z*r - t*u + u^2 + r^2, 5*y^2 - s^2 + s*a + a^2, x^2*z - 2*x^2*u + x^2*r + x*z^2 - x*z*u - x*z*r - 2*y^3 - t*u*v, 2*x*z*t + x*z*u - x*z*r + y^3 - w*t^2 - w*t*r - w*v*r + u^3 + u^2*r + u*v*r + v^2*r + s*a^2, x^2*w + x^2*r - x*z*u - x*z*r + y^3 + z^2*u - w*t*v + w*u^2 + t^2*u + u*v*r + v^2*r - s^2*a + s*a^2, x^2*z - x^2*u + x^2*r - x*z*w + x*z*v + z^3 + z^2*t + w*t^2 + t^2*u + t*u*v + t*u*r + u*r^2 + v*r^2 - a^3, x*z*w - x*z*u - x*z*v + x*z*r - y*a^2 + z^2*t - z^2*v + w*u^2 + w*r^2 + t^2*u + t*u*v, x^2*w + x^2*u - x^2*r + x*z*w + x*z*r + y*s*a + w*t^2 - w*t*v + w*t*r + w*u*r + u^2*v - u^2*r - u*v*r - s*a^2, x^2*z - x^2*u + x*z*t + x*z*u + x*z*v - y^3 + z^3 - z^2*u - z*w*v - z*t*v - w*t^2 - w*t*r - w*v*r + t*u^2 + u*v^2 + u*v*r + v^2*r + s^3 - s*a^2 - a^3]

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{3} y^{6} z^{5} + 2 x^{3} y^{5} z^{6} - x^{3} y^{4} z^{7} - 2 x^{3} y^{3} z^{8} + x^{3} y^{2} z^{9} + \cdots - z^{14} $

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

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 15.90.4.a.1 :

$\displaystyle X$	$=$	$\displaystyle -x+t$
$\displaystyle Y$	$=$	$\displaystyle -w+r$
$\displaystyle Z$	$=$	$\displaystyle x-z-w-t+u+2v$
$\displaystyle W$	$=$	$\displaystyle 2s-a$

Equation of the image curve:

$0$	$=$	$ 6X^{2}+XY+4Y^{2}-XZ+2YZ-Z^{2} $
	$=$	$ X^{2}Y+XY^{2}-Y^{3}+4XYZ+2Y^{2}Z+4YZ^{2}+W^{3} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 15.180.10.b.1 :

$\displaystyle X$	$=$	$\displaystyle a$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{5}u$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{5}r$

Equation of the image curve:

$0$

$=$

$ Y^{14}-13Y^{13}Z+59Y^{12}Z^{2}-112Y^{11}Z^{3}+126Y^{10}Z^{4}+X^{3}Y^{6}Z^{5}-239Y^{9}Z^{5}+2X^{3}Y^{5}Z^{6}+232Y^{8}Z^{6}-X^{3}Y^{4}Z^{7}+24Y^{7}Z^{7}-2X^{3}Y^{3}Z^{8}-232Y^{6}Z^{8}+X^{3}Y^{2}Z^{9}-239Y^{5}Z^{9}-126Y^{4}Z^{10}-112Y^{3}Z^{11}-59Y^{2}Z^{12}-13YZ^{13}-Z^{14} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{ns}}^+(3)$	$3$	$120$	$60$	$0$	$0$	full Jacobian
$X_{\mathrm{arith}}(5)$	$5$	$3$	$3$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{arith}}(5)$	$5$	$3$	$3$	$0$	$0$	full Jacobian
15.72.2-15.a.1.2	$15$	$5$	$5$	$2$	$0$	$1^{2}\cdot2^{3}$
15.72.2-15.a.2.1	$15$	$5$	$5$	$2$	$0$	$1^{2}\cdot2^{3}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
15.720.19-15.e.1.3	$15$	$2$	$2$	$19$	$2$	$1^{5}\cdot2^{2}$
15.720.19-15.g.1.6	$15$	$2$	$2$	$19$	$1$	$1^{5}\cdot2^{2}$
15.720.19-15.k.1.4	$15$	$2$	$2$	$19$	$1$	$1^{5}\cdot2^{2}$
15.720.19-15.l.1.2	$15$	$2$	$2$	$19$	$2$	$1^{5}\cdot2^{2}$
30.720.25-30.q.1.5	$30$	$2$	$2$	$25$	$2$	$1^{9}\cdot2^{3}$
30.720.25-30.by.1.7	$30$	$2$	$2$	$25$	$3$	$1^{9}\cdot2^{3}$
30.720.25-30.dz.1.3	$30$	$2$	$2$	$25$	$4$	$1^{7}\cdot2^{4}$
30.720.25-30.ea.1.7	$30$	$2$	$2$	$25$	$2$	$1^{7}\cdot2^{4}$
30.720.25-30.eq.1.6	$30$	$2$	$2$	$25$	$2$	$1^{7}\cdot2^{4}$
30.720.25-30.es.1.5	$30$	$2$	$2$	$25$	$3$	$1^{7}\cdot2^{4}$
30.720.25-30.eu.1.5	$30$	$2$	$2$	$25$	$4$	$1^{9}\cdot2^{3}$
30.720.25-30.ew.1.8	$30$	$2$	$2$	$25$	$2$	$1^{9}\cdot2^{3}$
30.1080.34-30.b.1.10	$30$	$3$	$3$	$34$	$3$	$1^{10}\cdot2^{7}$
45.1080.40-45.g.1.5	$45$	$3$	$3$	$40$	$9$	$1^{6}\cdot2^{4}\cdot4^{4}$
45.3240.118-45.k.1.4	$45$	$9$	$9$	$118$	$25$	$1^{15}\cdot2^{9}\cdot3^{3}\cdot4^{13}\cdot6\cdot8$
60.720.19-60.rg.1.10	$60$	$2$	$2$	$19$	$5$	$1^{5}\cdot2^{2}$
60.720.19-60.rm.1.10	$60$	$2$	$2$	$19$	$1$	$1^{5}\cdot2^{2}$
60.720.19-60.uo.1.10	$60$	$2$	$2$	$19$	$1$	$1^{5}\cdot2^{2}$
60.720.19-60.ur.1.12	$60$	$2$	$2$	$19$	$5$	$1^{5}\cdot2^{2}$
60.720.25-60.eb.1.21	$60$	$2$	$2$	$25$	$4$	$1^{9}\cdot2^{3}$
60.720.25-60.lb.1.13	$60$	$2$	$2$	$25$	$5$	$1^{9}\cdot2^{3}$
60.720.25-60.vi.1.14	$60$	$2$	$2$	$25$	$4$	$1^{7}\cdot2^{4}$
60.720.25-60.vl.1.10	$60$	$2$	$2$	$25$	$4$	$1^{7}\cdot2^{4}$
60.720.25-60.yl.1.10	$60$	$2$	$2$	$25$	$4$	$1^{7}\cdot2^{4}$
60.720.25-60.yr.1.10	$60$	$2$	$2$	$25$	$3$	$1^{7}\cdot2^{4}$
60.720.25-60.yx.1.9	$60$	$2$	$2$	$25$	$6$	$1^{9}\cdot2^{3}$
60.720.25-60.zd.1.14	$60$	$2$	$2$	$25$	$4$	$1^{9}\cdot2^{3}$
60.1440.55-60.bbt.1.20	$60$	$4$	$4$	$55$	$13$	$1^{21}\cdot2^{12}$

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