Modular curve 280.192.5-56.n.1.11

• Label: 280.192.5-56.n.1.11
• Level: $280$
• Index: $192$
• Genus: $5$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$280$		$\SL_2$-level:	$56$		Newform level:	$448$
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:	$\begin{bmatrix}5&182\\232&99\end{bmatrix}$, $\begin{bmatrix}31&196\\205&239\end{bmatrix}$, $\begin{bmatrix}99&112\\97&79\end{bmatrix}$, $\begin{bmatrix}151&168\\266&151\end{bmatrix}$, $\begin{bmatrix}277&210\\123&61\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.96.5.n.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 $	$=$	$ 2 x z + y w + y t - z w $
	$=$	$x^{2} - x w + 2 z^{2}$
	$=$	$2 x^{2} - 2 x w + 2 y^{2} - 2 z^{2} - 2 w^{2} - 2 w t - t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{8} - 8 x^{6} y^{2} - 4 x^{6} z^{2} + 38 x^{4} y^{4} + 4 x^{4} y^{2} z^{2} + 4 x^{4} z^{4} + \cdots + 4 y^{4} z^{4} $

Rational points

This modular curve has 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 $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{66691408896xw^{11}+82411186176xw^{10}t-227461063680xw^{9}t^{2}-802598897664xw^{8}t^{3}-1173614823936xw^{7}t^{4}-1067432297472xw^{6}t^{5}-664775027712xw^{5}t^{6}-292483699200xw^{4}t^{7}-88951262400xw^{3}t^{8}-17463600000xw^{2}t^{9}-1746360000xwt^{10}-331336829952yzw^{10}-1221636897792yzw^{9}t-1911058771968yzw^{8}t^{2}-1682256328704yzw^{7}t^{3}-923970977280yzw^{6}t^{4}-306365575680yzw^{5}t^{5}-39234972672yzw^{4}t^{6}+9788373504yzw^{3}t^{7}+5772513600yzw^{2}t^{8}+641390400yzwt^{9}+556240345280z^{2}w^{10}+2284876169280z^{2}w^{9}t+3958491017120z^{2}w^{8}t^{2}+3991433308160z^{2}w^{7}t^{3}+2624482694240z^{2}w^{6}t^{4}+1202630556960z^{2}w^{5}t^{5}+398764593360z^{2}w^{4}t^{6}+94443148800z^{2}w^{3}t^{7}+17709281100z^{2}w^{2}t^{8}+1967962500z^{2}wt^{9}+196796250z^{2}t^{10}+111592587552w^{12}+533364202272w^{11}t+1116665338000w^{10}t^{2}+1316832208192w^{9}t^{3}+900962874688w^{8}t^{4}+269069548176w^{7}t^{5}-109879643064w^{6}t^{6}-170209740960w^{5}t^{7}-98394086070w^{4}t^{8}-35179959150w^{3}t^{9}-8254997415w^{2}t^{10}-1291909500wt^{11}-107659125t^{12}}{680256xw^{11}+3482368xw^{10}t+6337408xw^{9}t^{2}+1624960xw^{8}t^{3}-13247616xw^{7}t^{4}-27282400xw^{6}t^{5}-28217072xw^{5}t^{6}-17905888xw^{4}t^{7}-7089236xw^{3}t^{8}-1617000xw^{2}t^{9}-161700xwt^{10}-1479104yzw^{10}-10837056yzw^{9}t-32714176yzw^{8}t^{2}-53356224yzw^{7}t^{3}-50703232yzw^{6}t^{4}-27294880yzw^{5}t^{5}-6581680yzw^{4}t^{6}+485072yzw^{3}t^{7}+534492yzw^{2}t^{8}+59388yzwt^{9}-2038016z^{2}w^{10}-17461056z^{2}w^{9}t-59744544z^{2}w^{8}t^{2}-112171392z^{2}w^{7}t^{3}-130750432z^{2}w^{6}t^{4}-100255680z^{2}w^{5}t^{5}-51700768z^{2}w^{4}t^{6}-17795232z^{2}w^{3}t^{7}-3973704z^{2}w^{2}t^{8}-583100z^{2}wt^{9}-58310z^{2}t^{10}-1261088w^{12}-8336224w^{11}t-24056368w^{10}t^{2}-38848832w^{9}t^{3}-36005728w^{8}t^{4}-13809840w^{7}t^{5}+9772744w^{6}t^{6}+18818688w^{5}t^{7}+14488950w^{4}t^{8}+6852650w^{3}t^{9}+2088821w^{2}t^{10}+382788wt^{11}+31899t^{12}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.96.5.n.1 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{2}t$

Equation of the image curve:

$0$

$=$

$ X^{8}-8X^{6}Y^{2}-4X^{6}Z^{2}+38X^{4}Y^{4}+4X^{4}Y^{2}Z^{2}+4X^{4}Z^{4}-88X^{2}Y^{6}+20X^{2}Y^{4}Z^{2}+8X^{2}Y^{2}Z^{4}+121Y^{8}+12Y^{6}Z^{2}+4Y^{4}Z^{4} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(7)$	$7$	$24$	$12$	$0$	$0$
40.24.0-8.h.1.1	$40$	$8$	$8$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.24.0-8.h.1.1	$40$	$8$	$8$	$0$	$0$
280.96.2-28.b.1.12	$280$	$2$	$2$	$2$	$?$
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.cn.1.5	$280$	$2$	$2$	$11$
280.384.11-56.cn.1.8	$280$	$2$	$2$	$11$
280.384.11-56.cn.2.6	$280$	$2$	$2$	$11$
280.384.11-56.cn.2.7	$280$	$2$	$2$	$11$
280.384.11-56.co.1.1	$280$	$2$	$2$	$11$
280.384.11-56.co.1.7	$280$	$2$	$2$	$11$
280.384.11-56.co.2.3	$280$	$2$	$2$	$11$
280.384.11-56.co.2.5	$280$	$2$	$2$	$11$
280.384.11-280.dx.1.8	$280$	$2$	$2$	$11$
280.384.11-280.dx.1.11	$280$	$2$	$2$	$11$
280.384.11-280.dx.2.4	$280$	$2$	$2$	$11$
280.384.11-280.dx.2.15	$280$	$2$	$2$	$11$
280.384.11-280.dy.1.8	$280$	$2$	$2$	$11$
280.384.11-280.dy.1.13	$280$	$2$	$2$	$11$
280.384.11-280.dy.2.6	$280$	$2$	$2$	$11$
280.384.11-280.dy.2.15	$280$	$2$	$2$	$11$