Modular curve 280.192.1-56.bs.2.4

• Label: 280.192.1-56.bs.2.4
• Level: $280$
• Index: $192$
• Genus: $1$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$280$		$\SL_2$-level:	$8$		Newform level:	$3136$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)		Cusp widths	$4^{8}\cdot8^{8}$		Cusp orbits	$4^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 96$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

8K1

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}51&16\\174&45\end{bmatrix}$, $\begin{bmatrix}65&104\\54&25\end{bmatrix}$, $\begin{bmatrix}89&252\\30&221\end{bmatrix}$, $\begin{bmatrix}279&124\\226&139\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.96.1.bs.2 for the level structure with $-I$)
Cyclic 280-isogeny field degree:	$96$
Cyclic 280-torsion field degree:	$9216$
Full 280-torsion field degree:	$7741440$

Jacobian

Conductor:	$?$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	3136.2.a.m

Models

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

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

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{4} + 4 x^{3} z - 3 x^{2} y^{2} + 14 x^{2} z^{2} - 6 x y^{2} z + 20 x z^{3} + 4 y^{4} - 15 y^{2} z^{2} + 18 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{1}{2^4}\cdot\frac{17741905920xz^{23}+204031918080xz^{22}w+1317336514560xz^{21}w^{2}+5976804556800xz^{20}w^{3}+20996436787200xz^{19}w^{4}+59950454538240xz^{18}w^{5}+143334362972160xz^{17}w^{6}+292396727992320xz^{16}w^{7}+515321503875072xz^{15}w^{8}+790962520227840xz^{14}w^{9}+1062384665739264xz^{13}w^{10}+1251288083275776xz^{12}w^{11}+1291805664694272xz^{11}w^{12}+1165524391139328xz^{10}w^{13}+913660661928960xz^{9}w^{14}+616450853942784xz^{8}w^{15}+352889713399680xz^{7}w^{16}+167748334157376xz^{6}w^{17}+64053383265504xz^{5}w^{18}+18603309666960xz^{4}w^{19}+3723097012488xz^{3}w^{20}+408695538204xz^{2}w^{21}+9360733698xzw^{22}-139062465xw^{23}-8111783936z^{24}-97341407232z^{23}w-661489975296z^{22}w^{2}-3171827056640z^{21}w^{3}-11822468431872z^{20}w^{4}-35942676037632z^{19}w^{5}-91829339029504z^{18}w^{6}-200917209710592z^{17}w^{7}-381276335308800z^{16}w^{8}-632808940765184z^{15}w^{9}-923344922394624z^{14}w^{10}-1187482168147968z^{13}w^{11}-1346253354700800z^{12}w^{12}-1342408275664896z^{11}w^{13}-1171393186022400z^{10}w^{14}-886937513960448z^{9}w^{15}-574970323089792z^{8}w^{16}-312554378423808z^{7}w^{17}-137756245900128z^{6}w^{18}-46349124375840z^{5}w^{19}-10470739604328z^{4}w^{20}-988369136624z^{3}w^{21}+146794148742z^{2}w^{22}+18102594126zw^{23}+273536188w^{24}}{w^{8}(3047424xz^{15}+22855680xz^{14}w+101326848xz^{13}w^{2}+311980032xz^{12}w^{3}+735381504xz^{11}w^{4}+1375435776xz^{10}w^{5}+2092485120xz^{9}w^{6}+2613904128xz^{8}w^{7}+2691685248xz^{7}w^{8}+2275022400xz^{6}w^{9}+1562331744xz^{5}w^{10}+854652240xz^{4}w^{11}+360869976xz^{3}w^{12}+111257172xz^{2}w^{13}+22564974xzw^{14}+2292705xw^{15}-2916352z^{16}-23330816z^{15}w-111583232z^{14}w^{2}-372793344z^{13}w^{3}-960620544z^{12}w^{4}-1978961920z^{11}w^{5}-3345920512z^{10}w^{6}-4693770752z^{9}w^{7}-5498077568z^{8}w^{8}-5371456000z^{7}w^{9}-4354352096z^{6}w^{10}-2893290144z^{5}w^{11}-1546838904z^{4}w^{12}-644718480z^{3}w^{13}-199248550z^{2}w^{14}-41236302zw^{15}-4509756w^{16})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.96.1.bs.2 :

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

Equation of the image curve:

$0$

$=$

$ X^{4}-3X^{2}Y^{2}+4Y^{4}+4X^{3}Z-6XY^{2}Z+14X^{2}Z^{2}-15Y^{2}Z^{2}+20XZ^{3}+18Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
40.96.0-8.g.1.4	$40$	$2$	$2$	$0$	$0$	full Jacobian
280.96.0-8.g.1.8	$280$	$2$	$2$	$0$	$?$	full Jacobian
280.96.0-56.g.1.1	$280$	$2$	$2$	$0$	$?$	full Jacobian
280.96.0-56.g.1.15	$280$	$2$	$2$	$0$	$?$	full Jacobian
280.96.0-56.h.1.1	$280$	$2$	$2$	$0$	$?$	full Jacobian
280.96.0-56.h.1.13	$280$	$2$	$2$	$0$	$?$	full Jacobian
280.96.0-56.x.2.3	$280$	$2$	$2$	$0$	$?$	full Jacobian
280.96.0-56.x.2.10	$280$	$2$	$2$	$0$	$?$	full Jacobian
280.96.1-56.bg.1.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1-56.bg.1.11	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1-56.bh.1.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1-56.bh.1.12	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1-56.bs.1.2	$280$	$2$	$2$	$1$	$?$	dimension zero
280.96.1-56.bs.1.14	$280$	$2$	$2$	$1$	$?$	dimension zero