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Properties

Label 16.192.5.o.2
Level $16$
Index $192$
Genus $5$
Analytic rank $0$
Cusps $24$
$\Q$-cusps $4$

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Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $128$
Index: $192$ $\PSL_2$-index:$192$
Genus: $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps: $24$ (of which $4$ are rational) Cusp widths $4^{8}\cdot8^{12}\cdot16^{4}$ Cusp orbits $1^{4}\cdot2^{2}\cdot4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16O5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.192.5.190

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}3&8\\12&15\end{bmatrix}$, $\begin{bmatrix}5&6\\12&7\end{bmatrix}$, $\begin{bmatrix}9&10\\8&7\end{bmatrix}$, $\begin{bmatrix}11&4\\4&7\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $C_2^3.C_2^4$
Contains $-I$: yes
Quadratic refinements: 16.384.5-16.o.2.1, 16.384.5-16.o.2.2, 16.384.5-16.o.2.3, 16.384.5-16.o.2.4, 16.384.5-16.o.2.5, 16.384.5-16.o.2.6, 16.384.5-16.o.2.7, 16.384.5-16.o.2.8, 48.384.5-16.o.2.1, 48.384.5-16.o.2.2, 48.384.5-16.o.2.3, 48.384.5-16.o.2.4, 48.384.5-16.o.2.5, 48.384.5-16.o.2.6, 48.384.5-16.o.2.7, 48.384.5-16.o.2.8, 80.384.5-16.o.2.1, 80.384.5-16.o.2.2, 80.384.5-16.o.2.3, 80.384.5-16.o.2.4, 80.384.5-16.o.2.5, 80.384.5-16.o.2.6, 80.384.5-16.o.2.7, 80.384.5-16.o.2.8, 112.384.5-16.o.2.1, 112.384.5-16.o.2.2, 112.384.5-16.o.2.3, 112.384.5-16.o.2.4, 112.384.5-16.o.2.5, 112.384.5-16.o.2.6, 112.384.5-16.o.2.7, 112.384.5-16.o.2.8, 176.384.5-16.o.2.1, 176.384.5-16.o.2.2, 176.384.5-16.o.2.3, 176.384.5-16.o.2.4, 176.384.5-16.o.2.5, 176.384.5-16.o.2.6, 176.384.5-16.o.2.7, 176.384.5-16.o.2.8, 208.384.5-16.o.2.1, 208.384.5-16.o.2.2, 208.384.5-16.o.2.3, 208.384.5-16.o.2.4, 208.384.5-16.o.2.5, 208.384.5-16.o.2.6, 208.384.5-16.o.2.7, 208.384.5-16.o.2.8, 240.384.5-16.o.2.1, 240.384.5-16.o.2.2, 240.384.5-16.o.2.3, 240.384.5-16.o.2.4, 240.384.5-16.o.2.5, 240.384.5-16.o.2.6, 240.384.5-16.o.2.7, 240.384.5-16.o.2.8, 272.384.5-16.o.2.1, 272.384.5-16.o.2.2, 272.384.5-16.o.2.3, 272.384.5-16.o.2.4, 272.384.5-16.o.2.5, 272.384.5-16.o.2.6, 272.384.5-16.o.2.7, 272.384.5-16.o.2.8, 304.384.5-16.o.2.1, 304.384.5-16.o.2.2, 304.384.5-16.o.2.3, 304.384.5-16.o.2.4, 304.384.5-16.o.2.5, 304.384.5-16.o.2.6, 304.384.5-16.o.2.7, 304.384.5-16.o.2.8
Cyclic 16-isogeny field degree: $4$
Cyclic 16-torsion field degree: $8$
Full 16-torsion field degree: $128$

Jacobian

Conductor: $2^{28}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2^{2}$
Newforms: 16.2.e.a, 64.2.a.a, 128.2.e.b

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $ $=$ $ y w + z t $
$=$ $y^{2} + 2 z w - t^{2}$
$=$ $4 x^{2} - y^{2} - z^{2} - w^{2} - t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} y^{4} - 4 x^{2} y^{6} - 4 x^{2} y^{4} z^{2} - 4 x^{2} y^{2} z^{4} - 4 x^{2} z^{6} + y^{8} + \cdots + z^{8} $
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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.

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

Maps between models of this curve

Birational map from canonical model to plane model:

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

Maps to other modular curves

$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2\,\frac{6144yz^{22}t-23364736yz^{18}t^{5}+1771206144yz^{14}t^{9}-2322865776yz^{10}t^{13}+123713912yz^{6}t^{17}-55553505yz^{2}t^{21}-1024z^{24}+384000z^{21}wt^{2}+1621504z^{20}t^{4}-150138240z^{17}wt^{6}-959198016z^{16}t^{8}+5593837248z^{13}wt^{10}+2796918624z^{12}t^{12}-2330491632z^{9}wt^{14}-675736268z^{8}t^{16}+504918156z^{5}wt^{18}-142746714z^{4}t^{20}-5221857zwt^{22}-1024w^{24}+3072w^{20}t^{4}-48768w^{16}t^{8}+137728w^{12}t^{12}-858732w^{8}t^{16}+2195484w^{4}t^{20}-4194304t^{24}}{t^{2}(4864yz^{18}t^{3}+214528yz^{14}t^{7}-988080yz^{10}t^{11}+10304yz^{6}t^{15}+1317yz^{2}t^{19}+1024z^{21}w+4608z^{20}t^{2}+72448z^{17}wt^{4}-66816z^{16}t^{6}-471808z^{13}wt^{8}-235904z^{12}t^{10}+1052464z^{9}wt^{12}-549848z^{8}t^{14}+6828z^{5}wt^{16}-2186z^{4}t^{18}+1317zwt^{20}-128w^{16}t^{6}+384w^{12}t^{10}-208w^{8}t^{14}-448w^{4}t^{18})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.1.f.2 $8$ $2$ $2$ $1$ $0$ $2^{2}$
16.96.2.b.1 $16$ $2$ $2$ $2$ $0$ $1\cdot2$
16.96.2.d.1 $16$ $2$ $2$ $2$ $0$ $1\cdot2$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
16.384.13.h.1 $16$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
16.384.13.i.2 $16$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
16.384.13.ba.1 $16$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
16.384.13.bb.1 $16$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.384.13.cs.1 $48$ $2$ $2$ $13$ $2$ $1^{2}\cdot2^{3}$
48.384.13.ct.1 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.384.13.ey.2 $48$ $2$ $2$ $13$ $2$ $1^{2}\cdot2^{3}$
48.384.13.ez.2 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.576.37.bgm.2 $48$ $3$ $3$ $37$ $1$ $1^{8}\cdot2^{4}\cdot8^{2}$
48.768.41.no.1 $48$ $4$ $4$ $41$ $1$ $1^{8}\cdot2^{6}\cdot8^{2}$
80.384.13.fu.2 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.fv.2 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.iu.2 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.iv.1 $80$ $2$ $2$ $13$ $?$ not computed
112.384.13.ck.1 $112$ $2$ $2$ $13$ $?$ not computed
112.384.13.cl.1 $112$ $2$ $2$ $13$ $?$ not computed
112.384.13.eq.2 $112$ $2$ $2$ $13$ $?$ not computed
112.384.13.er.2 $112$ $2$ $2$ $13$ $?$ not computed
176.384.13.ck.1 $176$ $2$ $2$ $13$ $?$ not computed
176.384.13.cl.1 $176$ $2$ $2$ $13$ $?$ not computed
176.384.13.eq.2 $176$ $2$ $2$ $13$ $?$ not computed
176.384.13.er.2 $176$ $2$ $2$ $13$ $?$ not computed
208.384.13.ec.2 $208$ $2$ $2$ $13$ $?$ not computed
208.384.13.ed.2 $208$ $2$ $2$ $13$ $?$ not computed
208.384.13.km.2 $208$ $2$ $2$ $13$ $?$ not computed
208.384.13.kn.1 $208$ $2$ $2$ $13$ $?$ not computed
240.384.13.bcs.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bct.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bma.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bmb.2 $240$ $2$ $2$ $13$ $?$ not computed
272.384.13.ck.2 $272$ $2$ $2$ $13$ $?$ not computed
272.384.13.cl.2 $272$ $2$ $2$ $13$ $?$ not computed
272.384.13.me.2 $272$ $2$ $2$ $13$ $?$ not computed
272.384.13.mf.1 $272$ $2$ $2$ $13$ $?$ not computed
304.384.13.ck.1 $304$ $2$ $2$ $13$ $?$ not computed
304.384.13.cl.1 $304$ $2$ $2$ $13$ $?$ not computed
304.384.13.eq.2 $304$ $2$ $2$ $13$ $?$ not computed
304.384.13.er.2 $304$ $2$ $2$ $13$ $?$ not computed