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Properties

Label 16.96.1.n.1
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 $2^{4}\cdot4^{2}$
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, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.196

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&14\\0&9\end{bmatrix}$, $\begin{bmatrix}3&11\\0&9\end{bmatrix}$, $\begin{bmatrix}5&0\\8&1\end{bmatrix}$, $\begin{bmatrix}7&12\\0&15\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $C_2\times C_4^2.Q_8$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.n.1.1, 16.192.1-16.n.1.2, 16.192.1-16.n.1.3, 16.192.1-16.n.1.4, 16.192.1-16.n.1.5, 16.192.1-16.n.1.6, 48.192.1-16.n.1.1, 48.192.1-16.n.1.2, 48.192.1-16.n.1.3, 48.192.1-16.n.1.4, 48.192.1-16.n.1.5, 48.192.1-16.n.1.6, 80.192.1-16.n.1.1, 80.192.1-16.n.1.2, 80.192.1-16.n.1.3, 80.192.1-16.n.1.4, 80.192.1-16.n.1.5, 80.192.1-16.n.1.6, 112.192.1-16.n.1.1, 112.192.1-16.n.1.2, 112.192.1-16.n.1.3, 112.192.1-16.n.1.4, 112.192.1-16.n.1.5, 112.192.1-16.n.1.6, 176.192.1-16.n.1.1, 176.192.1-16.n.1.2, 176.192.1-16.n.1.3, 176.192.1-16.n.1.4, 176.192.1-16.n.1.5, 176.192.1-16.n.1.6, 208.192.1-16.n.1.1, 208.192.1-16.n.1.2, 208.192.1-16.n.1.3, 208.192.1-16.n.1.4, 208.192.1-16.n.1.5, 208.192.1-16.n.1.6, 240.192.1-16.n.1.1, 240.192.1-16.n.1.2, 240.192.1-16.n.1.3, 240.192.1-16.n.1.4, 240.192.1-16.n.1.5, 240.192.1-16.n.1.6, 272.192.1-16.n.1.1, 272.192.1-16.n.1.2, 272.192.1-16.n.1.3, 272.192.1-16.n.1.4, 272.192.1-16.n.1.5, 272.192.1-16.n.1.6, 304.192.1-16.n.1.1, 304.192.1-16.n.1.2, 304.192.1-16.n.1.3, 304.192.1-16.n.1.4, 304.192.1-16.n.1.5, 304.192.1-16.n.1.6
Cyclic 16-isogeny field degree: $2$
Cyclic 16-torsion field degree: $16$
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 $ $=$ $ y^{2} + z^{2} - w^{2} $
$=$ $8 x^{2} + y^{2} + w^{2}$
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^2\,\frac{(z^{8}+56z^{6}w^{2}-40z^{4}w^{4}-32z^{2}w^{6}+16w^{8})^{3}}{w^{2}z^{4}(z-w)(z+w)(z^{2}-2w^{2})^{8}}$

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.n.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.k.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.ba.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.bb.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.g.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.s.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.t.2 $16$ $2$ $2$ $1$ $0$ 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.ca.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.cb.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hv.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hw.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.288.17.oa.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.pq.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
80.192.5.mx.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.my.1 $80$ $2$ $2$ $5$ $?$ not computed
112.192.5.hv.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.hw.1 $112$ $2$ $2$ $5$ $?$ not computed
176.192.5.hv.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.hw.1 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.mx.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.my.1 $208$ $2$ $2$ $5$ $?$ not computed
240.192.5.bzf.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bzg.1 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.mx.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.my.1 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.hv.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.hw.1 $304$ $2$ $2$ $5$ $?$ not computed