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Properties

Label 16.96.1.p.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 $2^{4}\cdot4^{2}$
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: 16.96.1.275

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&1\\0&13\end{bmatrix}$, $\begin{bmatrix}3&9\\0&9\end{bmatrix}$, $\begin{bmatrix}7&3\\0&11\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $C_2^4.\SD_{16}$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.p.2.1, 16.192.1-16.p.2.2, 16.192.1-16.p.2.3, 16.192.1-16.p.2.4, 32.192.1-16.p.2.1, 32.192.1-16.p.2.2, 48.192.1-16.p.2.1, 48.192.1-16.p.2.2, 48.192.1-16.p.2.3, 48.192.1-16.p.2.4, 80.192.1-16.p.2.1, 80.192.1-16.p.2.2, 80.192.1-16.p.2.3, 80.192.1-16.p.2.4, 96.192.1-16.p.2.1, 96.192.1-16.p.2.2, 112.192.1-16.p.2.1, 112.192.1-16.p.2.2, 112.192.1-16.p.2.3, 112.192.1-16.p.2.4, 160.192.1-16.p.2.1, 160.192.1-16.p.2.2, 176.192.1-16.p.2.1, 176.192.1-16.p.2.2, 176.192.1-16.p.2.3, 176.192.1-16.p.2.4, 208.192.1-16.p.2.1, 208.192.1-16.p.2.2, 208.192.1-16.p.2.3, 208.192.1-16.p.2.4, 224.192.1-16.p.2.1, 224.192.1-16.p.2.2, 240.192.1-16.p.2.1, 240.192.1-16.p.2.2, 240.192.1-16.p.2.3, 240.192.1-16.p.2.4, 272.192.1-16.p.2.1, 272.192.1-16.p.2.2, 272.192.1-16.p.2.3, 272.192.1-16.p.2.4, 304.192.1-16.p.2.1, 304.192.1-16.p.2.2, 304.192.1-16.p.2.3, 304.192.1-16.p.2.4
Cyclic 16-isogeny field degree: $1$
Cyclic 16-torsion field degree: $8$
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 $ $=$ $ 4 x^{2} + y^{2} + z^{2} $
$=$ $2 y^{2} + 4 z^{2} + w^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{(16z^{8}-224z^{6}w^{2}-40z^{4}w^{4}+8z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{4}(2z^{2}+w^{2})^{8}(4z^{2}+w^{2})}$

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.o.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.l.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.u.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.w.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.i.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.q.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.s.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
32.192.5.y.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.ba.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.288.17.oo.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.pz.1 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
96.192.5.ca.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ce.2 $96$ $2$ $2$ $5$ $?$ not computed
160.192.5.dm.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.dq.2 $160$ $2$ $2$ $5$ $?$ not computed
224.192.5.ca.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.ce.2 $224$ $2$ $2$ $5$ $?$ not computed