Modular curve 60.96.1-60.bn.1.4

• Label: 60.96.1-60.bn.1.4
• Level: $60$
• Index: $96$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$12$		Newform level:	$144$
Index:	$96$		$\PSL_2$-index:	$48$
Genus:	$1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$		Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.96.1.455

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}7&18\\41&23\end{bmatrix}$, $\begin{bmatrix}29&2\\54&25\end{bmatrix}$, $\begin{bmatrix}45&46\\2&47\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.48.1.bn.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree:	$12$
Cyclic 60-torsion field degree:	$96$
Full 60-torsion field degree:	$23040$

Jacobian

Conductor:	$2^{4}\cdot3^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	144.2.a.b

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $	$=$	$ 3 y^{2} - 3 y z - 3 z^{2} - w^{2} $
	$=$	$5 x^{2} - 2 y^{2} - 3 y z - 3 z^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{4} - 3 x^{2} y^{2} + 15 x^{2} z^{2} + y^{4} - 60 y^{2} z^{2} + 900 z^{4} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -2^4\cdot5^2\,\frac{14760427500xyz^{10}+6695743500xyz^{8}w^{2}+821137500xyz^{6}w^{4}-68388300xyz^{4}w^{6}-22651632xyz^{2}w^{8}-919212xyw^{10}+9121612500xz^{11}+6338776500xz^{9}w^{2}+1359301500xz^{7}w^{4}+34599420xz^{5}w^{6}-25642116xz^{3}w^{8}-2766252xzw^{10}-23947650000yz^{11}-12456787500yz^{9}w^{2}-2433827250yz^{7}w^{4}-366951600yz^{5}w^{6}-51805125yz^{3}w^{8}-2842650yzw^{10}-14800978125z^{12}-11268517500z^{10}w^{2}-3123400500z^{8}w^{4}-498480750z^{6}w^{6}-74390625z^{4}w^{8}-7470675z^{2}w^{10}-195725w^{12}}{w^{2}(51637500xyz^{8}+9618750xyz^{6}w^{2}-1113750xyz^{4}w^{4}-150300xyz^{2}w^{6}-1050xyw^{8}+31893750xz^{9}+13668750xz^{7}w^{2}+243000xz^{5}w^{4}-288900xz^{3}w^{6}-10800xzw^{8}+83531250yz^{9}+20098125yz^{7}w^{2}+3685500yz^{5}w^{4}+399150yz^{3}w^{6}+8130yzw^{8}+51637500z^{10}+24856875z^{8}w^{2}+4471875z^{6}w^{4}+695475z^{4}w^{6}+39480z^{2}w^{8}+223w^{10})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.1.bn.1 :

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle x$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{15}w$

Equation of the image curve:

$0$

$=$

$ X^{4}-3X^{2}Y^{2}+Y^{4}+15X^{2}Z^{2}-60Y^{2}Z^{2}+900Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
12.48.1-12.j.1.8	$12$	$2$	$2$	$1$	$0$	dimension zero
60.48.0-60.q.1.7	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.48.0-60.q.1.16	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.48.0-60.r.1.6	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.48.0-60.r.1.16	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.48.1-12.j.1.1	$60$	$2$	$2$	$1$	$0$	dimension zero

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.288.5-60.hd.1.1	$60$	$3$	$3$	$5$	$0$	$1^{4}$
60.480.17-60.mj.1.4	$60$	$5$	$5$	$17$	$0$	$1^{16}$
60.576.17-60.hz.1.8	$60$	$6$	$6$	$17$	$5$	$1^{16}$
60.960.33-60.om.1.15	$60$	$10$	$10$	$33$	$1$	$1^{32}$
180.288.5-180.bn.1.5	$180$	$3$	$3$	$5$	$?$	not computed
180.288.9-180.dv.1.8	$180$	$3$	$3$	$9$	$?$	not computed
180.288.9-180.er.1.8	$180$	$3$	$3$	$9$	$?$	not computed