↑

Properties

Label 16.48.1.i.1
Level $16$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

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: $8$ (of which $2$ are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16E1
Rouse and Zureick-Brown (RZB) label: X331
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.83

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&11\\0&13\end{bmatrix}$, $\begin{bmatrix}3&13\\0&7\end{bmatrix}$, $\begin{bmatrix}7&12\\0&15\end{bmatrix}$, $\begin{bmatrix}15&13\\0&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.i.1.1, 16.96.1-16.i.1.2, 16.96.1-16.i.1.3, 16.96.1-16.i.1.4, 16.96.1-16.i.1.5, 16.96.1-16.i.1.6, 16.96.1-16.i.1.7, 16.96.1-16.i.1.8, 32.96.1-16.i.1.1, 32.96.1-16.i.1.2, 32.96.1-16.i.1.3, 32.96.1-16.i.1.4, 48.96.1-16.i.1.1, 48.96.1-16.i.1.2, 48.96.1-16.i.1.3, 48.96.1-16.i.1.4, 48.96.1-16.i.1.5, 48.96.1-16.i.1.6, 48.96.1-16.i.1.7, 48.96.1-16.i.1.8, 80.96.1-16.i.1.1, 80.96.1-16.i.1.2, 80.96.1-16.i.1.3, 80.96.1-16.i.1.4, 80.96.1-16.i.1.5, 80.96.1-16.i.1.6, 80.96.1-16.i.1.7, 80.96.1-16.i.1.8, 96.96.1-16.i.1.1, 96.96.1-16.i.1.2, 96.96.1-16.i.1.3, 96.96.1-16.i.1.4, 112.96.1-16.i.1.1, 112.96.1-16.i.1.2, 112.96.1-16.i.1.3, 112.96.1-16.i.1.4, 112.96.1-16.i.1.5, 112.96.1-16.i.1.6, 112.96.1-16.i.1.7, 112.96.1-16.i.1.8, 160.96.1-16.i.1.1, 160.96.1-16.i.1.2, 160.96.1-16.i.1.3, 160.96.1-16.i.1.4, 176.96.1-16.i.1.1, 176.96.1-16.i.1.2, 176.96.1-16.i.1.3, 176.96.1-16.i.1.4, 176.96.1-16.i.1.5, 176.96.1-16.i.1.6, 176.96.1-16.i.1.7, 176.96.1-16.i.1.8, 208.96.1-16.i.1.1, 208.96.1-16.i.1.2, 208.96.1-16.i.1.3, 208.96.1-16.i.1.4, 208.96.1-16.i.1.5, 208.96.1-16.i.1.6, 208.96.1-16.i.1.7, 208.96.1-16.i.1.8, 224.96.1-16.i.1.1, 224.96.1-16.i.1.2, 224.96.1-16.i.1.3, 224.96.1-16.i.1.4, 240.96.1-16.i.1.1, 240.96.1-16.i.1.2, 240.96.1-16.i.1.3, 240.96.1-16.i.1.4, 240.96.1-16.i.1.5, 240.96.1-16.i.1.6, 240.96.1-16.i.1.7, 240.96.1-16.i.1.8, 272.96.1-16.i.1.1, 272.96.1-16.i.1.2, 272.96.1-16.i.1.3, 272.96.1-16.i.1.4, 272.96.1-16.i.1.5, 272.96.1-16.i.1.6, 272.96.1-16.i.1.7, 272.96.1-16.i.1.8, 304.96.1-16.i.1.1, 304.96.1-16.i.1.2, 304.96.1-16.i.1.3, 304.96.1-16.i.1.4, 304.96.1-16.i.1.5, 304.96.1-16.i.1.6, 304.96.1-16.i.1.7, 304.96.1-16.i.1.8
Cyclic 16-isogeny field degree: $1$
Cyclic 16-torsion field degree: $8$
Full 16-torsion field degree: $512$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x $
 Copy content Toggle raw display

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2\,\frac{360x^{2}y^{14}+24497170x^{2}y^{12}z^{2}+634938480x^{2}y^{10}z^{4}-353746131x^{2}y^{8}z^{6}+136395000x^{2}y^{6}z^{8}-11401455x^{2}y^{4}z^{10}+369000x^{2}y^{2}z^{12}-4095x^{2}z^{14}+44812xy^{14}z+129950640xy^{12}z^{3}+414345759xy^{10}z^{5}-268279920xy^{8}z^{7}+34440976xy^{6}z^{9}-1474200xy^{4}z^{11}+20481xy^{2}z^{13}+y^{16}+2116800y^{14}z^{2}+389660452y^{12}z^{4}+269567640y^{10}z^{6}+1694012y^{8}z^{8}+3224160y^{6}z^{10}+28434y^{4}z^{12}+360y^{2}z^{14}+z^{16}}{y^{2}(x^{2}y^{12}-100x^{2}y^{10}z^{2}-954x^{2}y^{8}z^{4}-2172x^{2}y^{6}z^{6}+315x^{2}y^{4}z^{8}+3060x^{2}y^{2}z^{10}+1025x^{2}z^{12}-12xy^{12}z-114xy^{10}z^{3}-1136xy^{8}z^{5}-5574xy^{6}z^{7}-9228xy^{4}z^{9}-4095xy^{2}z^{11}+56y^{12}z^{2}+696y^{10}z^{4}+3327y^{8}z^{6}+6008y^{6}z^{8}+3132y^{4}z^{10}-12y^{2}z^{12}+z^{14})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

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
$X_0(16)$ $16$ $2$ $2$ $0$ $0$ full Jacobian
16.24.1.a.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.p.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.p.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.q.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.q.2 $16$ $2$ $2$ $1$ $0$ dimension zero
32.96.3.k.1 $32$ $2$ $2$ $3$ $0$ $1^{2}$
32.96.3.l.1 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.l.2 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.m.1 $32$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.1.br.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.br.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bs.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bs.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.bg.1 $48$ $3$ $3$ $9$ $1$ $1^{8}$
48.192.9.mm.1 $48$ $4$ $4$ $9$ $1$ $1^{8}$
80.96.1.br.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.br.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.bs.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.bs.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.240.17.q.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.bo.1 $80$ $6$ $6$ $17$ $?$ not computed
96.96.3.k.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.l.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.l.2 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.m.1 $96$ $2$ $2$ $3$ $?$ not computed
112.96.1.br.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.br.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.bs.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.bs.2 $112$ $2$ $2$ $1$ $?$ dimension zero
160.96.3.k.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.l.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.l.2 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.m.1 $160$ $2$ $2$ $3$ $?$ not computed
176.96.1.br.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.br.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.bs.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.bs.2 $176$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.br.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.br.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.bs.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.bs.2 $208$ $2$ $2$ $1$ $?$ dimension zero
224.96.3.k.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.l.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.l.2 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.m.1 $224$ $2$ $2$ $3$ $?$ not computed
240.96.1.ex.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ex.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ey.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ey.2 $240$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.br.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.br.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.bs.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.bs.2 $272$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.br.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.br.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.bs.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.bs.2 $304$ $2$ $2$ $1$ $?$ dimension zero