Modular curve 112.192.2-16.c.1.7

• Label: 112.192.2-16.c.1.7
• Level: $112$
• Index: $192$
• Genus: $2$
• Cusps: $14$
• $\Q$-cusps: $2$

Invariants

Level:	$112$		$\SL_2$-level:	$16$		Newform level:	$128$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$2 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 14 }{2}$
Cusps:	$14$ (of which $2$ are rational)		Cusp widths	$4^{8}\cdot8^{4}\cdot16^{2}$		Cusp orbits	$1^{2}\cdot2^{4}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

16I2

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}23&108\\56&99\end{bmatrix}$, $\begin{bmatrix}25&52\\16&15\end{bmatrix}$, $\begin{bmatrix}81&40\\44&9\end{bmatrix}$, $\begin{bmatrix}81&52\\64&19\end{bmatrix}$, $\begin{bmatrix}105&16\\24&55\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 16.96.2.c.1 for the level structure with $-I$)
Cyclic 112-isogeny field degree:	$32$
Cyclic 112-torsion field degree:	$768$
Full 112-torsion field degree:	$258048$

Models

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

$ 0 $	$=$	$ x z w - x w^{2} - y w^{2} $
	$=$	$x z^{2} - x z w - y z w$
	$=$	$x^{2} z - x^{2} w - x y w$
	$=$	$x y z - x y w - y^{2} w$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 2 x^{4} + x^{2} y^{2} + 2 x y^{2} z - y^{2} z^{2} - 2 z^{4} $

Weierstrass model Weierstrass model

$ y^{2} $

$=$

$ 2x^{5} - 4x^{4} - 4x^{2} - 2x $

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.

Embedded model

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

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^2}\cdot\frac{4194304x^{20}+18612224x^{18}w^{2}+153550848x^{16}w^{4}-3029467136x^{14}w^{6}+9634119680x^{12}w^{8}-7687176192x^{10}w^{10}+12993454080x^{8}w^{12}-74483589120x^{6}w^{14}+98156113920x^{4}w^{16}-82307460096x^{2}w^{18}+9062285312xy^{19}+258840377344xy^{17}w^{2}+1879693974528xy^{15}w^{4}+2166864538624xy^{13}w^{6}+790020834304xy^{11}w^{8}+1717003129856xy^{9}w^{10}-4306471447552xy^{7}w^{12}+15969875772416xy^{5}w^{14}-3257632887808xy^{3}w^{16}+26469268444160xyw^{18}+2782294016y^{20}+82173890048y^{18}w^{2}+654062638336y^{16}w^{4}+1048285360128y^{14}w^{6}-29780075520y^{12}w^{8}+2238915082240y^{10}w^{10}-7824608877056y^{8}w^{12}+24760263546880y^{6}w^{14}+1738812156928y^{4}w^{16}+45036817906176y^{2}w^{18}-2684936z^{20}+45102576z^{19}w-293611337z^{18}w^{2}+1114344046z^{17}w^{3}-3444035294z^{16}w^{4}+9078099456z^{15}w^{5}-19848799172z^{14}w^{6}+37819244104z^{13}w^{7}-58632663880z^{12}w^{8}+67773565792z^{11}w^{9}-25196778462z^{10}w^{10}-38694825292z^{9}w^{11}+172250458284z^{8}w^{12}+193349359616z^{7}w^{13}-172626939092z^{6}w^{14}+1342516549384z^{5}w^{15}-2744325801888z^{4}w^{16}+6395443813808z^{3}w^{17}+6885997838911z^{2}w^{18}+5327373775502zw^{19}+9301944217506w^{20}}{w^{4}(65536x^{12}w^{4}+552960x^{10}w^{6}+4676608x^{8}w^{8}-28272640x^{6}w^{10}+37421056x^{4}w^{12}-37931520x^{2}w^{14}+26361856xy^{11}w^{4}+1689088xy^{9}w^{6}+1680942848xy^{7}w^{8}-15044860672xy^{5}w^{10}+204190337792xy^{3}w^{12}-2839852403456xyw^{14}+24708096y^{12}w^{4}-139494528y^{10}w^{6}+2192699712y^{8}w^{8}-25354082304y^{6}w^{10}+340817804544y^{4}w^{12}-4747312294144y^{2}w^{14}+8037z^{16}-97304z^{15}w+700706z^{14}w^{2}-4083640z^{13}w^{3}+20164503z^{12}w^{4}-89089344z^{11}w^{5}+358697912z^{10}w^{6}-1345206952z^{9}w^{7}+4638655483z^{8}w^{8}-15239383976z^{7}w^{9}+44805568362z^{6}w^{10}-129266575768z^{5}w^{11}+274612652657z^{4}w^{12}-694643915520z^{3}w^{13}-755137453124z^{2}w^{14}-579356619784zw^{15}-989243966824w^{16})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.96.2.c.1 :

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

Equation of the image curve:

$0$

$=$

$ 2X^{4}+X^{2}Y^{2}+2XY^{2}Z-Y^{2}Z^{2}-2Z^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 16.96.2.c.1 :

$\displaystyle X$	$=$	$\displaystyle \frac{1}{2}z+\frac{1}{2}w$
$\displaystyle Y$	$=$	$\displaystyle -\frac{1}{2}yz^{2}-yzw+\frac{1}{2}yw^{2}$
$\displaystyle Z$	$=$	$\displaystyle -\frac{1}{2}z+\frac{1}{2}w$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
56.96.0-8.c.1.4	$56$	$2$	$2$	$0$	$0$
112.96.0-8.c.1.3	$112$	$2$	$2$	$0$	$?$

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

Covering curve	Level	Index	Degree	Genus
112.384.5-16.i.1.7	$112$	$2$	$2$	$5$
112.384.5-16.i.2.6	$112$	$2$	$2$	$5$
112.384.5-16.k.2.11	$112$	$2$	$2$	$5$
112.384.5-16.k.5.10	$112$	$2$	$2$	$5$
112.384.5-16.u.1.7	$112$	$2$	$2$	$5$
112.384.5-16.u.2.10	$112$	$2$	$2$	$5$
112.384.5-16.v.1.8	$112$	$2$	$2$	$5$
112.384.5-16.v.2.5	$112$	$2$	$2$	$5$
112.384.5-112.co.1.14	$112$	$2$	$2$	$5$
112.384.5-112.co.2.14	$112$	$2$	$2$	$5$
112.384.5-112.cq.1.14	$112$	$2$	$2$	$5$
112.384.5-112.cq.2.14	$112$	$2$	$2$	$5$
112.384.5-112.db.1.12	$112$	$2$	$2$	$5$
112.384.5-112.db.2.14	$112$	$2$	$2$	$5$
112.384.5-112.dd.1.13	$112$	$2$	$2$	$5$
112.384.5-112.dd.2.11	$112$	$2$	$2$	$5$
112.384.7-16.g.1.1	$112$	$2$	$2$	$7$
112.384.7-16.h.1.2	$112$	$2$	$2$	$7$
112.384.7-16.o.1.10	$112$	$2$	$2$	$7$
112.384.7-16.q.1.7	$112$	$2$	$2$	$7$
112.384.7-112.w.1.2	$112$	$2$	$2$	$7$
112.384.7-112.x.1.1	$112$	$2$	$2$	$7$
112.384.7-112.bg.1.13	$112$	$2$	$2$	$7$
112.384.7-112.bh.1.14	$112$	$2$	$2$	$7$