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Properties

Label 112.192.2-16.d.1.7
Level $112$
Index $192$
Genus $2$
Cusps $14$
$\Q$-cusps $6$

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Invariants

Level: $112$ $\SL_2$-level: $16$ Newform level: $16$
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 $6$ are rational) Cusp widths $4^{8}\cdot8^{4}\cdot16^{2}$ Cusp orbits $1^{6}\cdot2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16I2

Level structure

$\GL_2(\Z/112\Z)$-generators: $\begin{bmatrix}1&16\\60&13\end{bmatrix}$, $\begin{bmatrix}33&28\\80&101\end{bmatrix}$, $\begin{bmatrix}47&64\\4&15\end{bmatrix}$, $\begin{bmatrix}47&80\\96&7\end{bmatrix}$, $\begin{bmatrix}95&40\\100&37\end{bmatrix}$
Contains $-I$: no $\quad$ (see 16.96.2.d.1 for the level structure with $-I$)
Cyclic 112-isogeny field degree: $32$
Cyclic 112-torsion field degree: $384$
Full 112-torsion field degree: $258048$

Models

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

$ 0 $ $=$ $ 2 x z w + y z w - y w^{2} $
$=$ $2 x z^{2} + y z^{2} - y z w$
$=$ $2 x y z + y^{2} z - y^{2} w$
$=$ $2 x^{2} z + x y z - x y w$
$=$$\cdots$
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Singular plane model Singular plane model

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

$ y^{2} + \left(x^{3} + x^{2} + x + 1\right) y $ $=$ $ -x^{4} - x^{3} - x^{2} - x $
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{524288x^{18}w^{2}+983040x^{16}w^{4}-1933312x^{14}w^{6}-1040384x^{12}w^{8}+2113536x^{10}w^{10}-2556672x^{8}w^{12}+178560x^{6}w^{14}+3726880x^{4}w^{16}-12225992x^{2}w^{18}+1098907648xy^{19}-801337344xy^{17}w^{2}-8653613056xy^{15}w^{4}+23705043968xy^{13}w^{6}-24442587648xy^{11}w^{8}-25214114816xy^{9}w^{10}+291160782592xy^{7}w^{12}-1599179515200xy^{5}w^{14}+6322782656032xy^{3}w^{16}-102614715600xyw^{18}+320864256y^{20}-769817600y^{18}w^{2}-1136867072y^{16}w^{4}+6115114240y^{14}w^{6}-1784763008y^{12}w^{8}-72669079040y^{10}w^{10}+531906369072y^{8}w^{12}-2882659210192y^{6}w^{14}+11587508823384y^{4}w^{16}-3295186675492y^{2}w^{18}-17z^{20}-338z^{19}w-3462z^{18}w^{2}-25088z^{17}w^{3}-159571z^{16}w^{4}-995842z^{15}w^{5}-6020924z^{14}w^{6}-33967536z^{13}w^{7}-179768681z^{12}w^{8}-912607850z^{11}w^{9}-4260393786z^{10}w^{10}-17719900750z^{9}w^{11}-64815639570z^{8}w^{12}-206113386266z^{7}w^{13}-554258845418z^{6}w^{14}-1181167852798z^{5}w^{15}-1702708042864z^{4}w^{16}-1012453753270z^{3}w^{17}-1157832223638z^{2}w^{18}-20523345810zw^{19}+51341929171w^{20}}{w^{4}(2048x^{10}w^{6}-4352x^{8}w^{8}+3712x^{6}w^{10}-5344x^{4}w^{12}+11520x^{2}w^{14}+131072xy^{11}w^{4}+7185664xy^{9}w^{6}-131464576xy^{7}w^{8}+1635288000xy^{5}w^{10}-17969505568xy^{3}w^{12}+185198083328xyw^{14}-491520y^{12}w^{4}+14769216y^{10}w^{6}-231754864y^{8}w^{8}+2859986128y^{6}w^{10}-31490952104y^{4}w^{12}+325091511936y^{2}w^{14}-17z^{16}-338z^{15}w-3530z^{14}w^{2}-26440z^{13}w^{3}-161077z^{12}w^{4}-850806z^{11}w^{5}-4046040z^{10}w^{6}-17753632z^{9}w^{7}-73101122z^{8}w^{8}-285889716z^{7}w^{9}-1070993212z^{6}w^{10}-3857790254z^{5}w^{11}-13264275473z^{4}w^{12}-41898186724z^{3}w^{13}-104861564030z^{2}w^{14}-38309357258zw^{15}-92599069123w^{16})}$

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

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

Equation of the image curve:

$0$ $=$ $ X^{2}Y^{2}+X^{3}Z-2XY^{2}Z-Y^{2}Z^{2}+XZ^{3} $

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

$\displaystyle X$ $=$ $\displaystyle -\frac{1}{2}z+\frac{1}{2}w$
$\displaystyle Y$ $=$ $\displaystyle -\frac{1}{4}yz^{2}+\frac{1}{2}yzw+\frac{1}{4}yw^{2}+\frac{1}{4}z^{3}+\frac{1}{4}zw^{2}$
$\displaystyle Z$ $=$ $\displaystyle -\frac{1}{2}z-\frac{1}{2}w$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
56.96.0-8.c.1.3 $56$ $2$ $2$ $0$ $0$
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
112.384.5-16.l.2.6 $112$ $2$ $2$ $5$
112.384.5-16.l.6.4 $112$ $2$ $2$ $5$
112.384.5-16.o.1.6 $112$ $2$ $2$ $5$
112.384.5-16.o.2.3 $112$ $2$ $2$ $5$
112.384.5-16.v.1.7 $112$ $2$ $2$ $5$
112.384.5-16.v.2.1 $112$ $2$ $2$ $5$
112.384.5-16.x.1.4 $112$ $2$ $2$ $5$
112.384.5-16.x.2.2 $112$ $2$ $2$ $5$
112.384.5-112.cs.1.15 $112$ $2$ $2$ $5$
112.384.5-112.cs.2.1 $112$ $2$ $2$ $5$
112.384.5-112.cu.1.14 $112$ $2$ $2$ $5$
112.384.5-112.cu.2.2 $112$ $2$ $2$ $5$
112.384.5-112.df.1.9 $112$ $2$ $2$ $5$
112.384.5-112.df.2.8 $112$ $2$ $2$ $5$
112.384.5-112.dh.1.9 $112$ $2$ $2$ $5$
112.384.5-112.dh.2.7 $112$ $2$ $2$ $5$
112.384.7-16.i.1.6 $112$ $2$ $2$ $7$
112.384.7-16.l.1.8 $112$ $2$ $2$ $7$
112.384.7-16.q.1.6 $112$ $2$ $2$ $7$
112.384.7-16.r.1.4 $112$ $2$ $2$ $7$
112.384.7-112.y.1.6 $112$ $2$ $2$ $7$
112.384.7-112.z.1.4 $112$ $2$ $2$ $7$
112.384.7-112.bi.1.10 $112$ $2$ $2$ $7$
112.384.7-112.bj.1.10 $112$ $2$ $2$ $7$