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Properties

Label 16.384.5-16.bx.2.2
Level $16$
Index $384$
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: $384$ $\PSL_2$-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 $2^{8}\cdot4^{4}\cdot8^{4}\cdot16^{8}$ Cusp orbits $1^{4}\cdot4^{5}$
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: 16N5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.384.5.534

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&3\\0&7\end{bmatrix}$, $\begin{bmatrix}1&9\\0&15\end{bmatrix}$, $\begin{bmatrix}9&3\\0&11\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $D_8:C_4$
Contains $-I$: no $\quad$ (see 16.192.5.bx.2 for the level structure with $-I$)
Cyclic 16-isogeny field degree: $1$
Cyclic 16-torsion field degree: $2$
Full 16-torsion field degree: $64$

Jacobian

Conductor: $2^{33}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2^{2}$
Newforms: 32.2.a.a, 128.2.e.a, 128.2.e.b

Models

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

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

$ 0 $ $=$ $ x^{4} y^{4} + 2 x^{4} y^{2} z^{2} + x^{4} z^{4} - 4 x^{2} y^{5} z - 24 x^{2} y^{3} z^{3} + \cdots + 4 y^{2} z^{6} $
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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
$(0:0:-1:0:1)$, $(0:-1:0:1:0)$, $(0:1:0:1:0)$, $(0:0:1:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{4095z^{2}w^{22}+81900z^{2}w^{21}t+700245z^{2}w^{20}t^{2}+3276000z^{2}w^{19}t^{3}+8809065z^{2}w^{18}t^{4}+12441060z^{2}w^{17}t^{5}+4318875z^{2}w^{16}t^{6}-10664640z^{2}w^{15}t^{7}-9791370z^{2}w^{14}t^{8}+4223160z^{2}w^{13}t^{9}+4437810z^{2}w^{12}t^{10}+4437810z^{2}w^{10}t^{12}-4223160z^{2}w^{9}t^{13}-9791370z^{2}w^{8}t^{14}+10664640z^{2}w^{7}t^{15}+4318875z^{2}w^{6}t^{16}-12441060z^{2}w^{5}t^{17}+8809065z^{2}w^{4}t^{18}-3276000z^{2}w^{3}t^{19}+700245z^{2}w^{2}t^{20}-81900z^{2}wt^{21}+4095z^{2}t^{22}-w^{24}-24w^{23}t-4347w^{22}t^{2}-67016w^{21}t^{3}-442839w^{20}t^{4}-1539792w^{19}t^{5}-2784293w^{18}t^{6}-1761696w^{17}t^{7}+1665270w^{16}t^{8}+1364848w^{15}t^{9}-4386990w^{14}t^{10}-1610736w^{13}t^{11}+17430322w^{12}t^{12}+15138816w^{11}t^{13}-35508810w^{10}t^{14}-25642528w^{9}t^{15}+58434075w^{8}t^{16}-4068504w^{7}t^{17}-48058343w^{6}t^{18}+45941112w^{5}t^{19}-21655659w^{4}t^{20}+6062096w^{3}t^{21}-1028097w^{2}t^{22}+98304wt^{23}-4096t^{24}}{t^{4}w^{4}(w^{2}+2wt-t^{2})^{2}(z^{2}w^{10}+8z^{2}w^{9}t+21z^{2}w^{8}t^{2}+16z^{2}w^{7}t^{3}+10z^{2}w^{6}t^{4}+10z^{2}w^{4}t^{6}-16z^{2}w^{3}t^{7}+21z^{2}w^{2}t^{8}-8z^{2}wt^{9}+z^{2}t^{10}+w^{12}+12w^{11}t+53w^{10}t^{2}+96w^{9}t^{3}+58w^{8}t^{4}+24w^{7}t^{5}+10w^{6}t^{6}+5w^{4}t^{8}-4w^{3}t^{9}+w^{2}t^{10})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 16.192.5.bx.2 :

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

Equation of the image curve:

$0$ $=$ $ X^{4}Y^{4}+2X^{4}Y^{2}Z^{2}+X^{4}Z^{4}-4X^{2}Y^{5}Z-24X^{2}Y^{3}Z^{3}-4X^{2}YZ^{5}+4Y^{6}Z^{2}+24Y^{4}Z^{4}+4Y^{2}Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.192.1-16.m.2.2 $16$ $2$ $2$ $1$ $0$ $2^{2}$
16.192.1-16.m.2.3 $16$ $2$ $2$ $1$ $0$ $2^{2}$
16.192.2-16.j.2.4 $16$ $2$ $2$ $2$ $0$ $1\cdot2$
16.192.2-16.j.2.7 $16$ $2$ $2$ $2$ $0$ $1\cdot2$
16.192.2-16.k.1.2 $16$ $2$ $2$ $2$ $0$ $1\cdot2$
16.192.2-16.k.1.3 $16$ $2$ $2$ $2$ $0$ $1\cdot2$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
16.768.13-16.bi.2.3 $16$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
16.768.13-16.bk.4.2 $16$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.13-32.bg.1.6 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.13-32.bl.1.3 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.13-32.cb.2.1 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.13-32.cg.2.2 $32$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
32.768.17-32.cb.1.2 $32$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{4}$
32.768.17-32.ce.1.1 $32$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{4}$
48.768.13-48.jl.3.5 $48$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{3}$
48.768.13-48.jn.3.1 $48$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{3}$
48.1152.37-48.fdc.1.2 $48$ $3$ $3$ $37$ $1$ $1^{8}\cdot2^{4}\cdot8^{2}$
48.1536.41-48.brl.1.2 $48$ $4$ $4$ $41$ $0$ $1^{8}\cdot2^{6}\cdot8^{2}$