Modular curve 280.192.6-56.d.2.2

• Label: 280.192.6-56.d.2.2
• Level: $280$
• Index: $192$
• Genus: $6$
• Cusps: $6$
• $\Q$-cusps: $2$

Invariants

Level:	$280$		$\SL_2$-level:	$56$		Newform level:	$448$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$6 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (of which $2$ are rational)		Cusp widths	$2^{2}\cdot8\cdot14^{2}\cdot56$		Cusp orbits	$1^{2}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 6$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

56D6

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}58&215\\95&204\end{bmatrix}$, $\begin{bmatrix}203&270\\110&43\end{bmatrix}$, $\begin{bmatrix}241&68\\216&179\end{bmatrix}$, $\begin{bmatrix}247&154\\226&81\end{bmatrix}$, $\begin{bmatrix}263&40\\72&197\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.96.6.d.2 for the level structure with $-I$)
Cyclic 280-isogeny field degree:	$24$
Cyclic 280-torsion field degree:	$2304$
Full 280-torsion field degree:	$7741440$

Models

Canonical model in $\mathbb{P}^{ 5 }$ defined by 9 equations

$ 0 $	$=$	$ y w - z t $
	$=$	$x w + y z$
	$=$	$x t + y^{2}$
	$=$	$3 x y - y u + w t$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

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

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

$(0:0:0:0:0:1)$, $(0:0:1:0:0:0)$

Maps to other modular curves

$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{3}{2^2}\cdot\frac{21364633280xt^{8}u+45469881216xt^{6}u^{3}-101042464436xt^{4}u^{5}-341836293884xt^{2}u^{7}-91663474686xu^{9}+4586471424z^{10}-6306398208z^{8}u^{2}+15497256960z^{6}u^{4}-9066147264z^{4}u^{6}+12749164614z^{2}u^{8}-1119465664zt^{9}-19179658928zt^{7}u^{2}+200184081644zt^{5}u^{4}-246756019589zt^{3}u^{6}+84332742194ztu^{8}+14930313728w^{2}t^{7}u-23518551376w^{2}t^{5}u^{3}-118462401528w^{2}t^{3}u^{5}-119913697559w^{2}tu^{7}-56405568t^{10}+4994113200t^{8}u^{2}-72292394124t^{6}u^{4}+74549340453t^{4}u^{6}+188010715809t^{2}u^{8}+32105299968u^{10}}{55552xt^{8}u+616056xt^{6}u^{3}+5246900xt^{4}u^{5}+4099733xt^{2}u^{7}-49707xu^{9}-13436928z^{6}u^{4}+9237888z^{4}u^{6}+1043199z^{2}u^{8}-152096zt^{9}-455896zt^{7}u^{2}-226610zt^{5}u^{4}-2719417zt^{3}u^{6}+776845ztu^{8}-736736w^{2}t^{7}u-638624w^{2}t^{5}u^{3}+3351342w^{2}t^{3}u^{5}-610108w^{2}tu^{7}+479136t^{10}+1576344t^{8}u^{2}+869058t^{6}u^{4}-1487793t^{4}u^{6}-1259643t^{2}u^{8}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.96.6.d.2 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle t$

Equation of the image curve:

$0$

$=$

$ 7X^{8}+7X^{4}Y^{2}Z^{2}-9X^{4}YZ^{3}+X^{4}Z^{4}-Y^{3}Z^{5}+Y^{2}Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
280.96.2-28.b.1.2	$280$	$2$	$2$	$2$	$?$
280.96.2-28.b.1.15	$280$	$2$	$2$	$2$	$?$

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

Covering curve	Level	Index	Degree	Genus
280.384.11-56.l.1.5	$280$	$2$	$2$	$11$
280.384.11-56.bu.2.6	$280$	$2$	$2$	$11$
280.384.11-56.cj.1.3	$280$	$2$	$2$	$11$
280.384.11-56.co.1.1	$280$	$2$	$2$	$11$
280.384.11-56.ed.2.3	$280$	$2$	$2$	$11$
280.384.11-56.ef.2.5	$280$	$2$	$2$	$11$
280.384.11-56.eh.2.3	$280$	$2$	$2$	$11$
280.384.11-56.ej.2.1	$280$	$2$	$2$	$11$
280.384.11-280.ip.1.11	$280$	$2$	$2$	$11$
280.384.11-280.ir.2.16	$280$	$2$	$2$	$11$
280.384.11-280.it.1.14	$280$	$2$	$2$	$11$
280.384.11-280.iv.1.15	$280$	$2$	$2$	$11$
280.384.11-280.mh.2.9	$280$	$2$	$2$	$11$
280.384.11-280.mj.2.14	$280$	$2$	$2$	$11$
280.384.11-280.ml.2.14	$280$	$2$	$2$	$11$
280.384.11-280.mn.2.15	$280$	$2$	$2$	$11$