Modular curve 112.192.3-8.i.1.1

• Label: 112.192.3-8.i.1.1
• Level: $112$
• Index: $192$
• Genus: $3$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$112$		$\SL_2$-level:	$16$		Newform level:	$64$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $4$ are rational)		Cusp widths	$8^{12}$		Cusp orbits	$1^{4}\cdot2^{2}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 3$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 3$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

8B3

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}39&12\\92&95\end{bmatrix}$, $\begin{bmatrix}41&20\\12&7\end{bmatrix}$, $\begin{bmatrix}73&80\\0&93\end{bmatrix}$, $\begin{bmatrix}89&64\\40&9\end{bmatrix}$, $\begin{bmatrix}89&96\\104&35\end{bmatrix}$, $\begin{bmatrix}103&8\\40&91\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 8.96.3.i.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 $\mathbb{P}^{4}$

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

Singular plane model Singular plane model

$ 0 $

$=$

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

Weierstrass model Weierstrass model

$ y^{2} $

$=$

$ x^{7} - 7x^{5} + 7x^{3} - x $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model

$(-1/2:0:0:-1:1)$, $(0:0:1:0:0)$, $(0:1:0:0:0)$, $(0:1:1:0:0)$

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{640xw^{13}-128xw^{12}t+4352xw^{11}t^{2}-256xw^{10}t^{3}+43904xw^{9}t^{4}-4480xw^{8}t^{5}+232960xw^{7}t^{6}-382152xw^{6}t^{7}+1223424xw^{5}t^{8}-2461408xw^{4}t^{9}+3658176xw^{3}t^{10}-5776040xw^{2}t^{11}+4838272xwt^{12}-640xt^{13}-y^{14}-6y^{12}t^{2}-22y^{10}t^{4}+24y^{8}t^{6}-1142y^{6}t^{8}+11232y^{4}t^{10}-124908y^{2}t^{12}+2yz^{13}+46yz^{11}t^{2}+56yz^{9}t^{4}+1240yz^{7}t^{6}+6794yz^{5}t^{8}+110102yz^{3}t^{10}+1209536yzt^{12}-20z^{12}t^{2}+16z^{10}t^{4}-888z^{8}t^{6}-8384z^{6}t^{8}-102988z^{4}t^{10}-1381280z^{2}t^{12}-224w^{14}+64w^{13}t-1568w^{12}t^{2}+384w^{11}t^{3}-15456w^{10}t^{4}+4032w^{9}t^{5}-83104w^{8}t^{6}+153428w^{7}t^{7}-430884w^{6}t^{8}+1026920w^{5}t^{9}-1319744w^{4}t^{10}+2783260w^{3}t^{11}-1862948w^{2}t^{12}+1631160wt^{13}-224t^{14}}{t^{4}(40xw^{9}-8xw^{8}t+272xw^{7}t^{2}-16xw^{6}t^{3}+904xw^{5}t^{4}+88xw^{4}t^{5}+2048xw^{3}t^{6}+184xw^{2}t^{7}+2368xwt^{8}-y^{6}t^{4}+10y^{4}t^{6}-118y^{2}t^{8}+2yz^{5}t^{4}-2yz^{3}t^{6}+592yzt^{8}-20z^{4}t^{6}-272z^{2}t^{8}-14w^{10}+4w^{9}t-98w^{8}t^{2}+24w^{7}t^{3}-322w^{6}t^{4}+68w^{5}t^{5}-686w^{4}t^{6}+236w^{3}t^{7}-732w^{2}t^{8}+508wt^{9})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.96.3.i.1 :

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

Equation of the image curve:

$0$

$=$

$ X^{3}Y^{2}+X^{4}Z-2X^{2}Y^{2}Z-5X^{3}Z^{2}-2XY^{2}Z^{2}+6X^{2}Z^{3}+4Y^{2}Z^{3}-2XZ^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 8.96.3.i.1 :

$\displaystyle X$	$=$	$\displaystyle -y+z$
$\displaystyle Y$	$=$	$\displaystyle -y^{3}t+2y^{2}zt+2yz^{2}t-4z^{3}t$
$\displaystyle Z$	$=$	$\displaystyle z$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.96.0-8.c.1.1	$112$	$2$	$2$	$0$	$?$
112.96.0-8.c.1.2	$112$	$2$	$2$	$0$	$?$

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

Covering curve	Level	Index	Degree	Genus
112.384.5-8.c.1.1	$112$	$2$	$2$	$5$
112.384.5-8.c.1.8	$112$	$2$	$2$	$5$
112.384.5-8.d.1.1	$112$	$2$	$2$	$5$
112.384.5-8.d.1.16	$112$	$2$	$2$	$5$
112.384.5-56.w.1.6	$112$	$2$	$2$	$5$
112.384.5-56.w.1.13	$112$	$2$	$2$	$5$
112.384.5-56.x.1.9	$112$	$2$	$2$	$5$
112.384.5-56.x.1.13	$112$	$2$	$2$	$5$
112.384.7-16.a.1.5	$112$	$2$	$2$	$7$
112.384.7-16.d.1.9	$112$	$2$	$2$	$7$
112.384.7-112.e.1.8	$112$	$2$	$2$	$7$
112.384.7-16.h.1.1	$112$	$2$	$2$	$7$
112.384.7-16.h.1.9	$112$	$2$	$2$	$7$
112.384.7-112.h.1.16	$112$	$2$	$2$	$7$
112.384.7-16.i.1.1	$112$	$2$	$2$	$7$
112.384.7-16.i.1.5	$112$	$2$	$2$	$7$
112.384.7-16.o.1.7	$112$	$2$	$2$	$7$
112.384.7-112.p.1.2	$112$	$2$	$2$	$7$
112.384.7-112.p.1.20	$112$	$2$	$2$	$7$
112.384.7-112.q.1.1	$112$	$2$	$2$	$7$
112.384.7-112.q.1.18	$112$	$2$	$2$	$7$
112.384.7-16.r.1.4	$112$	$2$	$2$	$7$
112.384.7-112.ba.1.16	$112$	$2$	$2$	$7$
112.384.7-112.bd.1.4	$112$	$2$	$2$	$7$
112.384.9-16.bo.1.2	$112$	$2$	$2$	$9$
112.384.9-16.bo.2.2	$112$	$2$	$2$	$9$
112.384.9-16.bp.1.2	$112$	$2$	$2$	$9$
112.384.9-16.bp.2.2	$112$	$2$	$2$	$9$
112.384.9-112.fy.1.11	$112$	$2$	$2$	$9$
112.384.9-112.fy.2.11	$112$	$2$	$2$	$9$
112.384.9-112.fz.1.11	$112$	$2$	$2$	$9$
112.384.9-112.fz.2.11	$112$	$2$	$2$	$9$
112.384.11-16.a.1.2	$112$	$2$	$2$	$11$
112.384.11-16.a.2.4	$112$	$2$	$2$	$11$
112.384.11-112.a.1.11	$112$	$2$	$2$	$11$
112.384.11-112.a.2.10	$112$	$2$	$2$	$11$
112.384.11-16.b.1.2	$112$	$2$	$2$	$11$
112.384.11-16.b.2.4	$112$	$2$	$2$	$11$
112.384.11-112.b.1.11	$112$	$2$	$2$	$11$
112.384.11-112.b.2.10	$112$	$2$	$2$	$11$