Modular curve 60.96.1-60.x.1.1

• Label: 60.96.1-60.x.1.1
• Level: $60$
• Index: $96$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$12$		Newform level:	$3600$
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:	60.96.1.235

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}35&28\\51&59\end{bmatrix}$, $\begin{bmatrix}43&12\\30&49\end{bmatrix}$, $\begin{bmatrix}49&14\\21&17\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.48.1.x.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree:	$12$
Cyclic 60-torsion field degree:	$192$
Full 60-torsion field degree:	$23040$

Jacobian

Conductor:	$2^{4}\cdot3^{2}\cdot5^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	3600.2.a.v

Models

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

$ 0 $	$=$	$ 7 x^{2} + x w - y^{2} - y w + z^{2} + w^{2} $
	$=$	$10 x y + 5 y^{2} - z^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 75 x^{4} - 15 x^{3} y + x^{2} y^{2} + 30 x^{2} z^{2} + x y z^{2} + 7 z^{4} $

Rational points

This modular curve has no real points, and therefore no 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 $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{2^4\cdot5^2}{3^3\cdot7^2}\cdot\frac{1419018328542150xz^{10}w+5385548807433669xz^{8}w^{3}+5197522096042920xz^{6}w^{5}+1384370751937250xz^{4}w^{7}+130857356661500xz^{2}w^{9}+3799803150000xw^{11}+60230279497500y^{2}z^{10}+165338949500235y^{2}z^{8}w^{2}+412299761705370y^{2}z^{6}w^{4}+24031225766100y^{2}z^{4}w^{6}-10558931202750y^{2}z^{2}w^{8}-1015856871250y^{2}w^{10}+790892026141770yz^{10}w+2658266763350163yz^{8}w^{3}+1867009406587500yz^{6}w^{5}+651811537742050yz^{4}w^{7}+86191614131000yz^{2}w^{9}+3931615317500yw^{11}-7413908629995z^{12}-322861025630430z^{10}w^{2}-220054875841941z^{8}w^{4}+373739063986050z^{6}w^{6}+13956283208350z^{4}w^{8}-12292069417750z^{2}w^{10}-942532756875w^{12}}{z^{2}(2416366400xz^{8}w-27328988050xz^{6}w^{3}+70765616250xz^{4}w^{5}+40698905625xz^{2}w^{7}-55759387500xw^{9}-113267175y^{2}z^{8}+2840245800y^{2}z^{6}w^{2}-13990057375y^{2}z^{4}w^{4}+29946105000y^{2}z^{2}w^{6}+2337356250y^{2}w^{8}+1736307160yz^{8}w-15349987450yz^{6}w^{3}+65162368250yz^{4}w^{5}-43958446875yz^{2}w^{7}-32554406250yw^{9}-26348574z^{10}+2316838090z^{8}w^{2}-12114943750z^{6}w^{4}-10416254625z^{4}w^{6}+49694461875z^{2}w^{8}+3116475000w^{10})}$

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

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle 10w$
$\displaystyle Z$	$=$	$\displaystyle z$

Equation of the image curve:

$0$

$=$

$ 75X^{4}-15X^{3}Y+X^{2}Y^{2}+30X^{2}Z^{2}+XYZ^{2}+7Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
12.48.0-12.j.1.3	$12$	$2$	$2$	$0$	$0$	full Jacobian
60.48.0-12.j.1.4	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.48.0-60.o.1.2	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.48.0-60.o.1.14	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.48.1-60.w.1.2	$60$	$2$	$2$	$1$	$0$	dimension zero
60.48.1-60.w.1.10	$60$	$2$	$2$	$1$	$0$	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.ir.1.4	$60$	$3$	$3$	$5$	$1$	$1^{4}$
60.480.17-60.ik.1.5	$60$	$5$	$5$	$17$	$8$	$1^{16}$
60.576.17-60.dp.1.2	$60$	$6$	$6$	$17$	$1$	$1^{16}$
60.960.33-60.gr.1.12	$60$	$10$	$10$	$33$	$11$	$1^{32}$
180.288.5-180.x.1.8	$180$	$3$	$3$	$5$	$?$	not computed
180.288.9-180.cs.1.5	$180$	$3$	$3$	$9$	$?$	not computed
180.288.9-180.cw.1.6	$180$	$3$	$3$	$9$	$?$	not computed