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Properties

Label 16.96.1.s.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: $32$
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 $2^{6}\cdot4$
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: X473
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.236

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&14\\8&15\end{bmatrix}$, $\begin{bmatrix}3&15\\0&9\end{bmatrix}$, $\begin{bmatrix}9&6\\0&1\end{bmatrix}$, $\begin{bmatrix}13&5\\8&1\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $\OD_{32}:C_2^3$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.s.2.1, 16.192.1-16.s.2.2, 16.192.1-16.s.2.3, 16.192.1-16.s.2.4, 16.192.1-16.s.2.5, 16.192.1-16.s.2.6, 32.192.1-16.s.2.1, 32.192.1-16.s.2.2, 32.192.1-16.s.2.3, 32.192.1-16.s.2.4, 48.192.1-16.s.2.1, 48.192.1-16.s.2.2, 48.192.1-16.s.2.3, 48.192.1-16.s.2.4, 48.192.1-16.s.2.5, 48.192.1-16.s.2.6, 80.192.1-16.s.2.1, 80.192.1-16.s.2.2, 80.192.1-16.s.2.3, 80.192.1-16.s.2.4, 80.192.1-16.s.2.5, 80.192.1-16.s.2.6, 96.192.1-16.s.2.1, 96.192.1-16.s.2.2, 96.192.1-16.s.2.3, 96.192.1-16.s.2.4, 112.192.1-16.s.2.1, 112.192.1-16.s.2.2, 112.192.1-16.s.2.3, 112.192.1-16.s.2.4, 112.192.1-16.s.2.5, 112.192.1-16.s.2.6, 160.192.1-16.s.2.1, 160.192.1-16.s.2.2, 160.192.1-16.s.2.3, 160.192.1-16.s.2.4, 176.192.1-16.s.2.1, 176.192.1-16.s.2.2, 176.192.1-16.s.2.3, 176.192.1-16.s.2.4, 176.192.1-16.s.2.5, 176.192.1-16.s.2.6, 208.192.1-16.s.2.1, 208.192.1-16.s.2.2, 208.192.1-16.s.2.3, 208.192.1-16.s.2.4, 208.192.1-16.s.2.5, 208.192.1-16.s.2.6, 224.192.1-16.s.2.1, 224.192.1-16.s.2.2, 224.192.1-16.s.2.3, 224.192.1-16.s.2.4, 240.192.1-16.s.2.1, 240.192.1-16.s.2.2, 240.192.1-16.s.2.3, 240.192.1-16.s.2.4, 240.192.1-16.s.2.5, 240.192.1-16.s.2.6, 272.192.1-16.s.2.1, 272.192.1-16.s.2.2, 272.192.1-16.s.2.3, 272.192.1-16.s.2.4, 272.192.1-16.s.2.5, 272.192.1-16.s.2.6, 304.192.1-16.s.2.1, 304.192.1-16.s.2.2, 304.192.1-16.s.2.3, 304.192.1-16.s.2.4, 304.192.1-16.s.2.5, 304.192.1-16.s.2.6
Cyclic 16-isogeny field degree: $2$
Cyclic 16-torsion field degree: $8$
Full 16-torsion field degree: $256$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

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

$ 0 $ $=$ $ x^{4} - 2 x^{2} y^{2} - 12 x^{2} z^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{3^{16}}\cdot\frac{525931019712yz^{22}w+17448204777120yz^{20}w^{3}-1986344744544yz^{18}w^{5}-135039533179248yz^{16}w^{7}+39319410531456yz^{14}w^{9}+234142912377792yz^{12}w^{11}+27060281478720yz^{10}w^{13}-136890066083040yz^{8}w^{15}-89883199990080yz^{6}w^{17}-23935903215840yz^{4}w^{19}-2965510133088yz^{2}w^{21}-141214768240yw^{23}-18391932981z^{24}-3886800750288z^{22}w^{2}-27945863642412z^{20}w^{4}+61060795493400z^{18}w^{6}+130289309349489z^{16}w^{8}-157569205524192z^{14}w^{10}-222792773014872z^{12}w^{12}+51702256914480z^{10}w^{14}+156201185513457z^{8}w^{16}+80045237798640z^{6}w^{18}+18710956791828z^{4}w^{20}+2118221523608z^{2}w^{22}+94143178827w^{24}}{z^{16}(12096yz^{6}w-4896yz^{4}w^{3}-16416yz^{2}w^{5}-3280yw^{7}-3456z^{8}-12528z^{6}w^{2}+14748z^{4}w^{4}+14216z^{2}w^{6}+2187w^{8})}$

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.p.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.l.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.z.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.bb.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.j.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.v.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.x.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.ch.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.ci.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.r.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bg.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.9.bq.2 $32$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
32.192.9.bs.1 $32$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
48.192.5.ih.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ii.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.qt.1 $48$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
48.384.17.qv.1 $48$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
80.192.5.nr.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ns.1 $80$ $2$ $2$ $5$ $?$ not computed
96.192.5.bj.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.cs.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.9.ea.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.ee.1 $96$ $2$ $2$ $9$ $?$ not computed
112.192.5.ih.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.ii.1 $112$ $2$ $2$ $5$ $?$ not computed
160.192.5.cj.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.ei.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.9.ee.2 $160$ $2$ $2$ $9$ $?$ not computed
160.192.9.ei.1 $160$ $2$ $2$ $9$ $?$ not computed
176.192.5.ih.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.ii.1 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.nr.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.ns.1 $208$ $2$ $2$ $5$ $?$ not computed
224.192.5.bj.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.cs.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.9.ea.2 $224$ $2$ $2$ $9$ $?$ not computed
224.192.9.ee.1 $224$ $2$ $2$ $9$ $?$ not computed
240.192.5.cap.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.caq.1 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.nr.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.ns.2 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.ih.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.ii.1 $304$ $2$ $2$ $5$ $?$ not computed