Modular curve 84.64.1-12.d.1.4

• Label: 84.64.1-12.d.1.4
• Level: $84$
• Index: $64$
• Genus: $1$
• Cusps: $4$
• $\Q$-cusps: $0$

Invariants

Level:	$84$		$\SL_2$-level:	$12$		Newform level:	$48$
Index:	$64$		$\PSL_2$-index:	$32$
Genus:	$1 = 1 + \frac{ 32 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (none of which are rational)		Cusp widths	$4^{2}\cdot12^{2}$		Cusp orbits	$2^{2}$
Elliptic points:	$0$ of order $2$ and $2$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 32$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12I1

Level structure

$\GL_2(\Z/84\Z)$-generators:	$\begin{bmatrix}1&42\\70&71\end{bmatrix}$, $\begin{bmatrix}50&9\\47&22\end{bmatrix}$, $\begin{bmatrix}66&79\\37&21\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 12.32.1.d.1 for the level structure with $-I$)
Cyclic 84-isogeny field degree:	$48$
Cyclic 84-torsion field degree:	$1152$
Full 84-torsion field degree:	$145152$

Jacobian

Conductor:	$?$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	48.2.a.a

Models

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

$ 0 $	$=$	$ 2 x^{2} + 2 x y + 3 y^{2} + y z + y w + z^{2} - z w + w^{2} $
	$=$	$x^{2} + 2 x y - 4 x z + 2 x w + y^{2} - y z - y w + z^{2} - z w + w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 11 x^{4} + 20 x^{3} z + x^{2} y^{2} + 5 x^{2} y z + 30 x^{2} z^{2} + x y^{2} z + 2 x y z^{2} + \cdots + 4 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 32 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 3^3\,\frac{(z-w)(30498408xz^{6}-21253056xz^{5}w-125379368xz^{4}w^{2}+224767040xz^{3}w^{3}-143213928xz^{2}w^{4}+37499648xzw^{5}-3001688xw^{6}-5306072y^{2}z^{5}+47080120y^{2}z^{4}w-88923152y^{2}z^{3}w^{2}+60131408y^{2}z^{2}w^{3}-12670744y^{2}zw^{4}-311560y^{2}w^{5}+4883104yz^{6}+27104400yz^{5}w-61719680yz^{4}w^{2}+5163552yz^{3}w^{3}+46230432yz^{2}w^{4}-24317488yzw^{5}+2572736yw^{6}-8301785z^{7}+26656415z^{6}w-33853149z^{5}w^{2}+18289083z^{4}w^{3}+5266773z^{3}w^{4}-13175619z^{2}w^{5}+5912609zw^{6}-711383w^{7})}{178992xz^{7}-2759520xz^{6}w+7160354xz^{5}w^{2}-7005616xz^{4}w^{3}+2398508xz^{3}w^{4}+350578xz^{2}w^{5}-374888xzw^{6}+55480xw^{7}+271829y^{2}z^{6}-1231407y^{2}z^{5}w+1448322y^{2}z^{4}w^{2}-99208y^{2}z^{3}w^{3}-673182y^{2}z^{2}w^{4}+316497y^{2}zw^{5}-32851y^{2}w^{6}+167648yz^{7}-1070123yz^{6}w+925073yz^{5}w^{2}+1206658yz^{4}w^{3}-1696092yz^{3}w^{4}+335089yz^{2}w^{5}+182037yzw^{6}-46402yw^{7}+75521z^{8}-31939z^{7}w-495337z^{6}w^{2}+1187313z^{5}w^{3}-1363137z^{4}w^{4}+837537z^{3}w^{5}-214793z^{2}w^{6}-4613zw^{7}+6289w^{8}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.32.1.d.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle 2w$
$\displaystyle Z$	$=$	$\displaystyle y$

Equation of the image curve:

$0$

$=$

$ 11X^{4}+X^{2}Y^{2}+20X^{3}Z+5X^{2}YZ+XY^{2}Z+30X^{2}Z^{2}+2XYZ^{2}+Y^{2}Z^{2}+16XZ^{3}+2YZ^{3}+4Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
84.16.0-12.b.1.4	$84$	$4$	$4$	$0$	$?$	full Jacobian
84.32.1-12.a.1.4	$84$	$2$	$2$	$1$	$?$	dimension zero
84.32.1-12.a.1.7	$84$	$2$	$2$	$1$	$?$	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
84.192.3-12.r.1.1	$84$	$3$	$3$	$3$	$?$	not computed
84.192.5-12.e.1.1	$84$	$3$	$3$	$5$	$?$	not computed
84.512.17-84.d.1.16	$84$	$8$	$8$	$17$	$?$	not computed
168.256.7-24.g.1.6	$168$	$4$	$4$	$7$	$?$	not computed
252.192.5-36.f.1.4	$252$	$3$	$3$	$5$	$?$	not computed
252.192.5-36.h.1.4	$252$	$3$	$3$	$5$	$?$	not computed
252.192.7-36.d.1.3	$252$	$3$	$3$	$7$	$?$	not computed