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Properties

Label 16.96.1.g.2
Level $16$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $64$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{8}\cdot4^{4}\cdot16^{4}$ Cusp orbits $2^{8}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16M1
Rouse and Zureick-Brown (RZB) label: X481
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.237

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&11\\0&13\end{bmatrix}$, $\begin{bmatrix}9&10\\0&3\end{bmatrix}$, $\begin{bmatrix}15&0\\0&5\end{bmatrix}$, $\begin{bmatrix}15&12\\0&7\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $\OD_{32}:C_2^3$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.g.2.1, 16.192.1-16.g.2.2, 16.192.1-16.g.2.3, 16.192.1-16.g.2.4, 16.192.1-16.g.2.5, 16.192.1-16.g.2.6, 32.192.1-16.g.2.1, 32.192.1-16.g.2.2, 32.192.1-16.g.2.3, 32.192.1-16.g.2.4, 48.192.1-16.g.2.1, 48.192.1-16.g.2.2, 48.192.1-16.g.2.3, 48.192.1-16.g.2.4, 48.192.1-16.g.2.5, 48.192.1-16.g.2.6, 80.192.1-16.g.2.1, 80.192.1-16.g.2.2, 80.192.1-16.g.2.3, 80.192.1-16.g.2.4, 80.192.1-16.g.2.5, 80.192.1-16.g.2.6, 96.192.1-16.g.2.1, 96.192.1-16.g.2.2, 96.192.1-16.g.2.3, 96.192.1-16.g.2.4, 112.192.1-16.g.2.1, 112.192.1-16.g.2.2, 112.192.1-16.g.2.3, 112.192.1-16.g.2.4, 112.192.1-16.g.2.5, 112.192.1-16.g.2.6, 160.192.1-16.g.2.1, 160.192.1-16.g.2.2, 160.192.1-16.g.2.3, 160.192.1-16.g.2.4, 176.192.1-16.g.2.1, 176.192.1-16.g.2.2, 176.192.1-16.g.2.3, 176.192.1-16.g.2.4, 176.192.1-16.g.2.5, 176.192.1-16.g.2.6, 208.192.1-16.g.2.1, 208.192.1-16.g.2.2, 208.192.1-16.g.2.3, 208.192.1-16.g.2.4, 208.192.1-16.g.2.5, 208.192.1-16.g.2.6, 224.192.1-16.g.2.1, 224.192.1-16.g.2.2, 224.192.1-16.g.2.3, 224.192.1-16.g.2.4, 240.192.1-16.g.2.1, 240.192.1-16.g.2.2, 240.192.1-16.g.2.3, 240.192.1-16.g.2.4, 240.192.1-16.g.2.5, 240.192.1-16.g.2.6, 272.192.1-16.g.2.1, 272.192.1-16.g.2.2, 272.192.1-16.g.2.3, 272.192.1-16.g.2.4, 272.192.1-16.g.2.5, 272.192.1-16.g.2.6, 304.192.1-16.g.2.1, 304.192.1-16.g.2.2, 304.192.1-16.g.2.3, 304.192.1-16.g.2.4, 304.192.1-16.g.2.5, 304.192.1-16.g.2.6
Cyclic 16-isogeny field degree: $1$
Cyclic 16-torsion field degree: $4$
Full 16-torsion field degree: $256$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ x^{2} + 2 y^{2} - 4 y z - 2 z^{2} $
$=$ $2 y z - 5 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 2 x^{2} y^{2} - 6 x^{2} z^{2} + z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{524288y^{24}+12582912y^{22}w^{2}+127401984y^{20}w^{4}+707264512y^{18}w^{6}+2341502976y^{16}w^{8}+4707287040y^{14}w^{10}+5592604672y^{12}w^{12}+3618852864y^{10}w^{14}+1050320640y^{8}w^{16}+65601920y^{6}w^{18}-6828768y^{4}w^{20}+188592y^{2}w^{22}-1862645149230957z^{24}+1788139343261718z^{22}w^{2}-679492950439446z^{20}w^{4}+126838684081998z^{18}w^{6}-11072158813404z^{16}w^{8}+160217285130z^{14}w^{10}+43304443182z^{12}w^{12}-2658691134z^{10}w^{14}-17761185z^{8}w^{16}+7929180z^{6}w^{18}-285948z^{4}w^{20}-7092z^{2}w^{22}-4426w^{24}}{w^{16}(8y^{8}+64y^{6}w^{2}+136y^{4}w^{4}+56y^{2}w^{6}-12207z^{8}+3906z^{6}w^{2}-234z^{4}w^{4}-6z^{2}w^{6}-3w^{8})}$

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
8.48.0.m.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.e.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.u.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.v.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.d.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.q.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.r.1 $16$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
16.192.5.bj.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.bk.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.f.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.h.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.i.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.j.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.fp.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fq.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.fk.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.jj.1 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
80.192.5.kf.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.kg.2 $80$ $2$ $2$ $5$ $?$ not computed
96.192.5.n.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.r.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.s.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.t.2 $96$ $2$ $2$ $5$ $?$ not computed
112.192.5.fp.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fq.2 $112$ $2$ $2$ $5$ $?$ not computed
160.192.5.r.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.v.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.w.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.bb.2 $160$ $2$ $2$ $5$ $?$ not computed
176.192.5.fp.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fq.2 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.kf.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.kg.2 $208$ $2$ $2$ $5$ $?$ not computed
224.192.5.n.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.r.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.s.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.t.2 $224$ $2$ $2$ $5$ $?$ not computed
240.192.5.bod.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.boe.2 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.kf.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.kg.2 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.fp.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fq.2 $304$ $2$ $2$ $5$ $?$ not computed