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Properties

Label 16.48.1.j.1
Level $16$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $32$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $2^{4}$
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: 16E1
Rouse and Zureick-Brown (RZB) label: X332
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.84

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}5&11\\8&15\end{bmatrix}$, $\begin{bmatrix}7&11\\8&11\end{bmatrix}$, $\begin{bmatrix}11&5\\8&1\end{bmatrix}$, $\begin{bmatrix}15&12\\0&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.j.1.1, 16.96.1-16.j.1.2, 16.96.1-16.j.1.3, 16.96.1-16.j.1.4, 16.96.1-16.j.1.5, 16.96.1-16.j.1.6, 16.96.1-16.j.1.7, 16.96.1-16.j.1.8, 32.96.1-16.j.1.1, 32.96.1-16.j.1.2, 32.96.1-16.j.1.3, 32.96.1-16.j.1.4, 48.96.1-16.j.1.1, 48.96.1-16.j.1.2, 48.96.1-16.j.1.3, 48.96.1-16.j.1.4, 48.96.1-16.j.1.5, 48.96.1-16.j.1.6, 48.96.1-16.j.1.7, 48.96.1-16.j.1.8, 80.96.1-16.j.1.1, 80.96.1-16.j.1.2, 80.96.1-16.j.1.3, 80.96.1-16.j.1.4, 80.96.1-16.j.1.5, 80.96.1-16.j.1.6, 80.96.1-16.j.1.7, 80.96.1-16.j.1.8, 96.96.1-16.j.1.1, 96.96.1-16.j.1.2, 96.96.1-16.j.1.3, 96.96.1-16.j.1.4, 112.96.1-16.j.1.1, 112.96.1-16.j.1.2, 112.96.1-16.j.1.3, 112.96.1-16.j.1.4, 112.96.1-16.j.1.5, 112.96.1-16.j.1.6, 112.96.1-16.j.1.7, 112.96.1-16.j.1.8, 160.96.1-16.j.1.1, 160.96.1-16.j.1.2, 160.96.1-16.j.1.3, 160.96.1-16.j.1.4, 176.96.1-16.j.1.1, 176.96.1-16.j.1.2, 176.96.1-16.j.1.3, 176.96.1-16.j.1.4, 176.96.1-16.j.1.5, 176.96.1-16.j.1.6, 176.96.1-16.j.1.7, 176.96.1-16.j.1.8, 208.96.1-16.j.1.1, 208.96.1-16.j.1.2, 208.96.1-16.j.1.3, 208.96.1-16.j.1.4, 208.96.1-16.j.1.5, 208.96.1-16.j.1.6, 208.96.1-16.j.1.7, 208.96.1-16.j.1.8, 224.96.1-16.j.1.1, 224.96.1-16.j.1.2, 224.96.1-16.j.1.3, 224.96.1-16.j.1.4, 240.96.1-16.j.1.1, 240.96.1-16.j.1.2, 240.96.1-16.j.1.3, 240.96.1-16.j.1.4, 240.96.1-16.j.1.5, 240.96.1-16.j.1.6, 240.96.1-16.j.1.7, 240.96.1-16.j.1.8, 272.96.1-16.j.1.1, 272.96.1-16.j.1.2, 272.96.1-16.j.1.3, 272.96.1-16.j.1.4, 272.96.1-16.j.1.5, 272.96.1-16.j.1.6, 272.96.1-16.j.1.7, 272.96.1-16.j.1.8, 304.96.1-16.j.1.1, 304.96.1-16.j.1.2, 304.96.1-16.j.1.3, 304.96.1-16.j.1.4, 304.96.1-16.j.1.5, 304.96.1-16.j.1.6, 304.96.1-16.j.1.7, 304.96.1-16.j.1.8
Cyclic 16-isogeny field degree: $2$
Cyclic 16-torsion field degree: $16$
Full 16-torsion field degree: $512$

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 $ $=$ $ 6 x^{2} - 2 x y + z^{2} $
$=$ $8 x^{2} + 14 x y - 2 y^{2} - z^{2} + w^{2}$
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Singular plane model Singular plane model

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

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{8}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{4}z$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 5\,\frac{4608y^{2}z^{8}w^{2}+26496y^{2}z^{4}w^{6}+1638y^{2}w^{10}+1280z^{12}-13824z^{10}w^{2}+55536z^{8}w^{4}-79488z^{6}w^{6}+27717z^{4}w^{8}-4914z^{2}w^{10}+461w^{12}}{w^{2}z^{4}(32y^{2}z^{4}+2y^{2}w^{4}-96z^{6}-41z^{4}w^{2}-6z^{2}w^{4}-w^{6})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.r.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.24.0.h.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.24.1.b.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
16.96.1.r.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.r.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.s.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.s.2 $16$ $2$ $2$ $1$ $0$ dimension zero
32.96.5.k.1 $32$ $2$ $2$ $5$ $0$ $1^{4}$
32.96.5.l.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.96.5.l.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.96.5.m.1 $32$ $2$ $2$ $5$ $2$ $1^{4}$
48.96.1.cb.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cb.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cc.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cc.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.bn.1 $48$ $3$ $3$ $9$ $3$ $1^{8}$
48.192.9.mr.1 $48$ $4$ $4$ $9$ $0$ $1^{8}$
80.96.1.cb.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.cb.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.cc.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.cc.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.240.17.t.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.bt.1 $80$ $6$ $6$ $17$ $?$ not computed
96.96.5.k.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.l.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.l.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.m.1 $96$ $2$ $2$ $5$ $?$ not computed
112.96.1.cb.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.cb.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.cc.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.cc.2 $112$ $2$ $2$ $1$ $?$ dimension zero
160.96.5.k.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.l.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.l.2 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.m.1 $160$ $2$ $2$ $5$ $?$ not computed
176.96.1.cb.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.cb.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.cc.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.cc.2 $176$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cb.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cb.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cc.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cc.2 $208$ $2$ $2$ $1$ $?$ dimension zero
224.96.5.k.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.l.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.l.2 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.m.1 $224$ $2$ $2$ $5$ $?$ not computed
240.96.1.gf.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gf.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gg.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gg.2 $240$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cb.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cb.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cc.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cc.2 $272$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cb.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cb.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cc.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cc.2 $304$ $2$ $2$ $1$ $?$ dimension zero