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Properties

Label 16.48.1.o.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: $64$
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: X334
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.93

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}3&11\\12&13\end{bmatrix}$, $\begin{bmatrix}15&2\\8&15\end{bmatrix}$, $\begin{bmatrix}15&6\\0&1\end{bmatrix}$, $\begin{bmatrix}15&6\\8&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.o.1.1, 16.96.1-16.o.1.2, 16.96.1-16.o.1.3, 16.96.1-16.o.1.4, 16.96.1-16.o.1.5, 16.96.1-16.o.1.6, 16.96.1-16.o.1.7, 16.96.1-16.o.1.8, 32.96.1-16.o.1.1, 32.96.1-16.o.1.2, 32.96.1-16.o.1.3, 32.96.1-16.o.1.4, 48.96.1-16.o.1.1, 48.96.1-16.o.1.2, 48.96.1-16.o.1.3, 48.96.1-16.o.1.4, 48.96.1-16.o.1.5, 48.96.1-16.o.1.6, 48.96.1-16.o.1.7, 48.96.1-16.o.1.8, 80.96.1-16.o.1.1, 80.96.1-16.o.1.2, 80.96.1-16.o.1.3, 80.96.1-16.o.1.4, 80.96.1-16.o.1.5, 80.96.1-16.o.1.6, 80.96.1-16.o.1.7, 80.96.1-16.o.1.8, 96.96.1-16.o.1.1, 96.96.1-16.o.1.2, 96.96.1-16.o.1.3, 96.96.1-16.o.1.4, 112.96.1-16.o.1.1, 112.96.1-16.o.1.2, 112.96.1-16.o.1.3, 112.96.1-16.o.1.4, 112.96.1-16.o.1.5, 112.96.1-16.o.1.6, 112.96.1-16.o.1.7, 112.96.1-16.o.1.8, 160.96.1-16.o.1.1, 160.96.1-16.o.1.2, 160.96.1-16.o.1.3, 160.96.1-16.o.1.4, 176.96.1-16.o.1.1, 176.96.1-16.o.1.2, 176.96.1-16.o.1.3, 176.96.1-16.o.1.4, 176.96.1-16.o.1.5, 176.96.1-16.o.1.6, 176.96.1-16.o.1.7, 176.96.1-16.o.1.8, 208.96.1-16.o.1.1, 208.96.1-16.o.1.2, 208.96.1-16.o.1.3, 208.96.1-16.o.1.4, 208.96.1-16.o.1.5, 208.96.1-16.o.1.6, 208.96.1-16.o.1.7, 208.96.1-16.o.1.8, 224.96.1-16.o.1.1, 224.96.1-16.o.1.2, 224.96.1-16.o.1.3, 224.96.1-16.o.1.4, 240.96.1-16.o.1.1, 240.96.1-16.o.1.2, 240.96.1-16.o.1.3, 240.96.1-16.o.1.4, 240.96.1-16.o.1.5, 240.96.1-16.o.1.6, 240.96.1-16.o.1.7, 240.96.1-16.o.1.8, 272.96.1-16.o.1.1, 272.96.1-16.o.1.2, 272.96.1-16.o.1.3, 272.96.1-16.o.1.4, 272.96.1-16.o.1.5, 272.96.1-16.o.1.6, 272.96.1-16.o.1.7, 272.96.1-16.o.1.8, 304.96.1-16.o.1.1, 304.96.1-16.o.1.2, 304.96.1-16.o.1.3, 304.96.1-16.o.1.4, 304.96.1-16.o.1.5, 304.96.1-16.o.1.6, 304.96.1-16.o.1.7, 304.96.1-16.o.1.8
Cyclic 16-isogeny field degree: $2$
Cyclic 16-torsion field degree: $16$
Full 16-torsion field degree: $512$

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 $ $=$ $ - 2 y w + z^{2} $
$=$ $x^{2} - 32 y^{2} - 2 w^{2}$
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Singular plane model Singular plane model

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

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{(16y^{4}+16y^{2}w^{2}+w^{4})^{3}}{w^{2}y^{8}(16y^{2}+w^{2})}$

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.y.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.24.0.i.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.24.1.c.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
16.96.3.cc.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.cd.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.dr.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
16.96.3.ds.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
32.96.3.o.1 $32$ $2$ $2$ $3$ $0$ $1^{2}$
32.96.3.p.1 $32$ $2$ $2$ $3$ $0$ $1^{2}$
32.96.5.p.1 $32$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
32.96.5.p.2 $32$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.3.ht.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.hu.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.jr.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.js.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.144.9.ce.1 $48$ $3$ $3$ $9$ $2$ $1^{8}$
48.192.9.ro.1 $48$ $4$ $4$ $9$ $1$ $1^{8}$
80.96.3.jn.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.jo.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.lt.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.lu.1 $80$ $2$ $2$ $3$ $?$ not computed
80.240.17.bc.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.dy.1 $80$ $6$ $6$ $17$ $?$ not computed
96.96.3.o.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.p.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.5.o.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.o.2 $96$ $2$ $2$ $5$ $?$ not computed
112.96.3.hl.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.hm.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.jj.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.jk.1 $112$ $2$ $2$ $3$ $?$ not computed
160.96.3.o.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.p.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.5.o.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.o.2 $160$ $2$ $2$ $5$ $?$ not computed
176.96.3.hl.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.hm.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.jj.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.jk.1 $176$ $2$ $2$ $3$ $?$ not computed
208.96.3.jn.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.jo.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.lt.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.lu.1 $208$ $2$ $2$ $3$ $?$ not computed
224.96.3.o.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.p.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.5.o.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.o.2 $224$ $2$ $2$ $5$ $?$ not computed
240.96.3.bbj.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bbk.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bgz.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bha.1 $240$ $2$ $2$ $3$ $?$ not computed
272.96.3.jf.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.jg.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.lt.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.lu.1 $272$ $2$ $2$ $3$ $?$ not computed
304.96.3.hl.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.hm.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.jj.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.jk.1 $304$ $2$ $2$ $3$ $?$ not computed