↑

Properties

Label 16.48.1.x.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: $32$
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: 16G1
Rouse and Zureick-Brown (RZB) label: X338
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.47

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&6\\8&11\end{bmatrix}$, $\begin{bmatrix}3&9\\8&11\end{bmatrix}$, $\begin{bmatrix}3&13\\8&15\end{bmatrix}$, $\begin{bmatrix}7&8\\8&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.x.1.1, 16.96.1-16.x.1.2, 16.96.1-16.x.1.3, 16.96.1-16.x.1.4, 16.96.1-16.x.1.5, 16.96.1-16.x.1.6, 16.96.1-16.x.1.7, 16.96.1-16.x.1.8, 32.96.1-16.x.1.1, 32.96.1-16.x.1.2, 32.96.1-16.x.1.3, 32.96.1-16.x.1.4, 48.96.1-16.x.1.1, 48.96.1-16.x.1.2, 48.96.1-16.x.1.3, 48.96.1-16.x.1.4, 48.96.1-16.x.1.5, 48.96.1-16.x.1.6, 48.96.1-16.x.1.7, 48.96.1-16.x.1.8, 80.96.1-16.x.1.1, 80.96.1-16.x.1.2, 80.96.1-16.x.1.3, 80.96.1-16.x.1.4, 80.96.1-16.x.1.5, 80.96.1-16.x.1.6, 80.96.1-16.x.1.7, 80.96.1-16.x.1.8, 96.96.1-16.x.1.1, 96.96.1-16.x.1.2, 96.96.1-16.x.1.3, 96.96.1-16.x.1.4, 112.96.1-16.x.1.1, 112.96.1-16.x.1.2, 112.96.1-16.x.1.3, 112.96.1-16.x.1.4, 112.96.1-16.x.1.5, 112.96.1-16.x.1.6, 112.96.1-16.x.1.7, 112.96.1-16.x.1.8, 160.96.1-16.x.1.1, 160.96.1-16.x.1.2, 160.96.1-16.x.1.3, 160.96.1-16.x.1.4, 176.96.1-16.x.1.1, 176.96.1-16.x.1.2, 176.96.1-16.x.1.3, 176.96.1-16.x.1.4, 176.96.1-16.x.1.5, 176.96.1-16.x.1.6, 176.96.1-16.x.1.7, 176.96.1-16.x.1.8, 208.96.1-16.x.1.1, 208.96.1-16.x.1.2, 208.96.1-16.x.1.3, 208.96.1-16.x.1.4, 208.96.1-16.x.1.5, 208.96.1-16.x.1.6, 208.96.1-16.x.1.7, 208.96.1-16.x.1.8, 224.96.1-16.x.1.1, 224.96.1-16.x.1.2, 224.96.1-16.x.1.3, 224.96.1-16.x.1.4, 240.96.1-16.x.1.1, 240.96.1-16.x.1.2, 240.96.1-16.x.1.3, 240.96.1-16.x.1.4, 240.96.1-16.x.1.5, 240.96.1-16.x.1.6, 240.96.1-16.x.1.7, 240.96.1-16.x.1.8, 272.96.1-16.x.1.1, 272.96.1-16.x.1.2, 272.96.1-16.x.1.3, 272.96.1-16.x.1.4, 272.96.1-16.x.1.5, 272.96.1-16.x.1.6, 272.96.1-16.x.1.7, 272.96.1-16.x.1.8, 304.96.1-16.x.1.1, 304.96.1-16.x.1.2, 304.96.1-16.x.1.3, 304.96.1-16.x.1.4, 304.96.1-16.x.1.5, 304.96.1-16.x.1.6, 304.96.1-16.x.1.7, 304.96.1-16.x.1.8
Cyclic 16-isogeny field degree: $2$
Cyclic 16-torsion field degree: $8$
Full 16-torsion field degree: $512$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 11x - 14 $
 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
$(-2:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{24x^{2}y^{14}+21586x^{2}y^{12}z^{2}+3441624x^{2}y^{10}z^{4}+193245609x^{2}y^{8}z^{6}+4400033856x^{2}y^{6}z^{8}+38490501129x^{2}y^{4}z^{10}+142771224564x^{2}y^{2}z^{12}+190919389185x^{2}z^{14}+316xy^{14}z+146472xy^{12}z^{3}+17836719xy^{10}z^{5}+861766122xy^{8}z^{7}+17968009096xy^{6}z^{9}+151627161576xy^{4}z^{11}+552379830297xy^{2}z^{13}+730920968190xz^{15}+y^{16}+2832y^{14}z^{2}+676068y^{12}z^{4}+51635064y^{10}z^{6}+1673064092y^{8}z^{8}+24815025696y^{6}z^{10}+173617528794y^{4}z^{12}+566431350864y^{2}z^{14}+698164379641z^{16}}{z^{5}y^{2}(307x^{2}y^{6}z+112528x^{2}y^{4}z^{3}+8659504x^{2}y^{2}z^{5}+175629440x^{2}z^{7}+xy^{8}+2854xy^{6}z^{2}+637168xy^{4}z^{4}+38479136xy^{2}z^{6}+672384512xz^{8}+24y^{8}z+18528y^{6}z^{3}+2153152y^{4}z^{5}+72453504y^{2}z^{7}+642251264z^{9})}$

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
$X_{\pm1}(8)$ $8$ $2$ $2$ $0$ $0$ full Jacobian
16.24.0.f.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.f.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.j.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.o.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.s.2 $16$ $2$ $2$ $1$ $0$ dimension zero
32.96.5.u.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.96.5.y.1 $32$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
32.96.5.bc.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.96.5.bg.1 $32$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.1.cq.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cu.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dg.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dk.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.fn.1 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bbg.1 $48$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
80.96.1.cp.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.ct.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.df.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.dj.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.240.17.cl.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.gh.1 $80$ $6$ $6$ $17$ $?$ not computed
96.96.5.bw.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.ca.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cu.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cy.1 $96$ $2$ $2$ $5$ $?$ not computed
112.96.1.cp.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.ct.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.df.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.dj.2 $112$ $2$ $2$ $1$ $?$ dimension zero
160.96.5.bw.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.ca.2 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.cu.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.cy.1 $160$ $2$ $2$ $5$ $?$ not computed
176.96.1.cp.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.ct.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.df.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.dj.2 $176$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cp.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.ct.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.df.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.dj.2 $208$ $2$ $2$ $1$ $?$ dimension zero
224.96.5.bw.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.ca.2 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.cu.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.cy.1 $224$ $2$ $2$ $5$ $?$ not computed
240.96.1.js.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ka.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ky.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.lg.2 $240$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cp.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.ct.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.df.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.dj.2 $272$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cp.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.ct.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.df.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.dj.2 $304$ $2$ $2$ $1$ $?$ dimension zero