Modular curve 280.192.5-28.e.1.5

• Label: 280.192.5-28.e.1.5
• Level: $280$
• Index: $192$
• Genus: $5$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$280$		$\SL_2$-level:	$56$		Newform level:	$112$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot14^{2}\cdot28^{2}$		Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

28E5

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}1&168\\150&169\end{bmatrix}$, $\begin{bmatrix}19&126\\217&167\end{bmatrix}$, $\begin{bmatrix}95&182\\83&27\end{bmatrix}$, $\begin{bmatrix}139&224\\168&255\end{bmatrix}$, $\begin{bmatrix}205&224\\116&65\end{bmatrix}$, $\begin{bmatrix}233&210\\257&69\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 28.96.5.e.1 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 $\mathbb{P}^{ 4 }$ defined by 3 equations

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

Singular plane model Singular plane model

$ 0 $

$=$

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

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=3$, and therefore no rational points.

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle -\frac{368944128y^{2}w^{10}+2097537840y^{2}w^{8}t^{2}+749927808y^{2}w^{6}t^{4}-9368352y^{2}w^{4}t^{6}-32935680y^{2}w^{2}t^{8}-5764752y^{2}t^{10}+2329791696yzw^{9}t+3497326848yzw^{7}t^{3}+50938272yzw^{5}t^{5}-46599552yzw^{3}t^{7}-36206064yzwt^{9}-2197105345z^{2}w^{10}-11156694915z^{2}w^{8}t^{2}-4273446730z^{2}w^{6}t^{4}-25952710z^{2}w^{4}t^{6}-11121285z^{2}w^{2}t^{8}-5549095z^{2}t^{10}+368947264w^{12}+4443945407w^{10}t^{2}+5798358293w^{8}t^{4}+799680854w^{6}t^{6}+367178w^{4}t^{8}-5512933w^{2}t^{10}+49t^{12}}{27y^{2}w^{8}t^{2}+889y^{2}w^{6}t^{4}+1869y^{2}w^{4}t^{6}+335y^{2}w^{2}t^{8}-27yzw^{9}t+2031yzw^{7}t^{3}-7469yzw^{5}t^{5}-3863yzw^{3}t^{7}-384yzwt^{9}-z^{2}w^{10}+250z^{2}w^{8}t^{2}-1890z^{2}w^{6}t^{4}-3045z^{2}w^{4}t^{6}-602z^{2}w^{2}t^{8}+64z^{2}t^{10}-w^{10}t^{2}+231w^{8}t^{4}-1547w^{6}t^{6}-714w^{4}t^{8}+15w^{2}t^{10}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 28.96.5.e.1 :

$\displaystyle X$	$=$	$\displaystyle x+w$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle z$

Equation of the image curve:

$0$

$=$

$ X^{4}Y^{4}+4X^{3}Y^{3}Z^{2}+X^{2}Y^{6}-5X^{2}Y^{4}Z^{2}+7X^{2}Y^{2}Z^{4}+2XY^{5}Z^{2}-10XY^{3}Z^{4}+6XYZ^{6}+Y^{6}Z^{2}+3Y^{4}Z^{4}-4Y^{2}Z^{6}+2Z^{8} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(7)$	$7$	$24$	$12$	$0$	$0$
40.24.0-4.c.1.2	$40$	$8$	$8$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.24.0-4.c.1.2	$40$	$8$	$8$	$0$	$0$
280.96.2-28.b.1.9	$280$	$2$	$2$	$2$	$?$
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
280.384.11-28.j.1.6	$280$	$2$	$2$	$11$
280.384.11-28.j.2.6	$280$	$2$	$2$	$11$
280.384.11-140.w.1.11	$280$	$2$	$2$	$11$
280.384.11-140.w.2.13	$280$	$2$	$2$	$11$
280.384.11-56.ci.1.1	$280$	$2$	$2$	$11$
280.384.11-56.ci.1.8	$280$	$2$	$2$	$11$
280.384.11-56.ci.2.2	$280$	$2$	$2$	$11$
280.384.11-56.ci.2.7	$280$	$2$	$2$	$11$
280.384.11-56.cj.1.3	$280$	$2$	$2$	$11$
280.384.11-56.cj.1.6	$280$	$2$	$2$	$11$
280.384.11-56.cj.2.4	$280$	$2$	$2$	$11$
280.384.11-56.cj.2.5	$280$	$2$	$2$	$11$
280.384.11-56.ck.1.12	$280$	$2$	$2$	$11$
280.384.11-56.ck.2.11	$280$	$2$	$2$	$11$
280.384.11-280.ds.1.8	$280$	$2$	$2$	$11$
280.384.11-280.ds.1.9	$280$	$2$	$2$	$11$
280.384.11-280.ds.2.4	$280$	$2$	$2$	$11$
280.384.11-280.ds.2.13	$280$	$2$	$2$	$11$
280.384.11-280.dt.1.6	$280$	$2$	$2$	$11$
280.384.11-280.dt.1.11	$280$	$2$	$2$	$11$
280.384.11-280.dt.2.2	$280$	$2$	$2$	$11$
280.384.11-280.dt.2.15	$280$	$2$	$2$	$11$
280.384.11-280.du.1.21	$280$	$2$	$2$	$11$
280.384.11-280.du.2.20	$280$	$2$	$2$	$11$
280.384.13-56.u.1.13	$280$	$2$	$2$	$13$
280.384.13-280.u.1.17	$280$	$2$	$2$	$13$
280.384.13-56.v.1.12	$280$	$2$	$2$	$13$
280.384.13-280.v.1.21	$280$	$2$	$2$	$13$
280.384.13-56.w.1.11	$280$	$2$	$2$	$13$
280.384.13-280.w.1.21	$280$	$2$	$2$	$13$
280.384.13-56.x.1.11	$280$	$2$	$2$	$13$
280.384.13-280.x.1.23	$280$	$2$	$2$	$13$