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Properties

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

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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: $8$ (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: X339
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.54

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}5&0\\0&9\end{bmatrix}$, $\begin{bmatrix}9&1\\0&3\end{bmatrix}$, $\begin{bmatrix}13&12\\8&7\end{bmatrix}$, $\begin{bmatrix}15&9\\0&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.x.2.1, 16.96.1-16.x.2.2, 16.96.1-16.x.2.3, 16.96.1-16.x.2.4, 16.96.1-16.x.2.5, 16.96.1-16.x.2.6, 16.96.1-16.x.2.7, 16.96.1-16.x.2.8, 32.96.1-16.x.2.1, 32.96.1-16.x.2.2, 32.96.1-16.x.2.3, 32.96.1-16.x.2.4, 48.96.1-16.x.2.1, 48.96.1-16.x.2.2, 48.96.1-16.x.2.3, 48.96.1-16.x.2.4, 48.96.1-16.x.2.5, 48.96.1-16.x.2.6, 48.96.1-16.x.2.7, 48.96.1-16.x.2.8, 80.96.1-16.x.2.1, 80.96.1-16.x.2.2, 80.96.1-16.x.2.3, 80.96.1-16.x.2.4, 80.96.1-16.x.2.5, 80.96.1-16.x.2.6, 80.96.1-16.x.2.7, 80.96.1-16.x.2.8, 96.96.1-16.x.2.1, 96.96.1-16.x.2.2, 96.96.1-16.x.2.3, 96.96.1-16.x.2.4, 112.96.1-16.x.2.1, 112.96.1-16.x.2.2, 112.96.1-16.x.2.3, 112.96.1-16.x.2.4, 112.96.1-16.x.2.5, 112.96.1-16.x.2.6, 112.96.1-16.x.2.7, 112.96.1-16.x.2.8, 160.96.1-16.x.2.1, 160.96.1-16.x.2.2, 160.96.1-16.x.2.3, 160.96.1-16.x.2.4, 176.96.1-16.x.2.1, 176.96.1-16.x.2.2, 176.96.1-16.x.2.3, 176.96.1-16.x.2.4, 176.96.1-16.x.2.5, 176.96.1-16.x.2.6, 176.96.1-16.x.2.7, 176.96.1-16.x.2.8, 208.96.1-16.x.2.1, 208.96.1-16.x.2.2, 208.96.1-16.x.2.3, 208.96.1-16.x.2.4, 208.96.1-16.x.2.5, 208.96.1-16.x.2.6, 208.96.1-16.x.2.7, 208.96.1-16.x.2.8, 224.96.1-16.x.2.1, 224.96.1-16.x.2.2, 224.96.1-16.x.2.3, 224.96.1-16.x.2.4, 240.96.1-16.x.2.1, 240.96.1-16.x.2.2, 240.96.1-16.x.2.3, 240.96.1-16.x.2.4, 240.96.1-16.x.2.5, 240.96.1-16.x.2.6, 240.96.1-16.x.2.7, 240.96.1-16.x.2.8, 272.96.1-16.x.2.1, 272.96.1-16.x.2.2, 272.96.1-16.x.2.3, 272.96.1-16.x.2.4, 272.96.1-16.x.2.5, 272.96.1-16.x.2.6, 272.96.1-16.x.2.7, 272.96.1-16.x.2.8, 304.96.1-16.x.2.1, 304.96.1-16.x.2.2, 304.96.1-16.x.2.3, 304.96.1-16.x.2.4, 304.96.1-16.x.2.5, 304.96.1-16.x.2.6, 304.96.1-16.x.2.7, 304.96.1-16.x.2.8
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

Weierstrass model Weierstrass model

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

$\displaystyle j$ $=$ $\displaystyle \frac{744x^{2}y^{14}+861743506x^{2}y^{12}z^{2}+3946501303224x^{2}y^{10}z^{4}+2404052725370889x^{2}y^{8}z^{6}+450301688638460016x^{2}y^{6}z^{8}+34266169928476438809x^{2}y^{4}z^{10}+1121366330130233819844x^{2}y^{2}z^{12}+13119880523481218023425x^{2}z^{14}+196876xy^{14}z+20032110312xy^{12}z^{3}+40645158196479xy^{10}z^{5}+16765017079652202xy^{8}z^{7}+2481637326273847576xy^{6}z^{9}+160992383698961342616xy^{4}z^{11}+4690996378190219311737xy^{2}z^{13}+50228506469524739981310xz^{15}+y^{16}+21481872y^{14}z^{2}+317842308708y^{12}z^{4}+319718459371944y^{10}z^{6}+82473813056611532y^{8}z^{8}+8224738401779468256y^{6}z^{10}+365602252934307716394y^{4}z^{12}+7147543060259635457184y^{2}z^{14}+47977490845124607868921z^{16}}{y^{2}(x^{2}y^{12}+4x^{2}y^{10}z^{2}-14x^{2}y^{8}z^{4}-16x^{2}y^{6}z^{6}+171x^{2}y^{4}z^{8}-28x^{2}y^{2}z^{10}+x^{2}z^{12}+14xy^{10}z^{3}+100xy^{8}z^{5}+106xy^{6}z^{7}-364xy^{4}z^{9}+57xy^{2}z^{11}-2xz^{13}-16y^{12}z^{2}-112y^{10}z^{4}-177y^{8}z^{6}-200y^{6}z^{8}-1108y^{4}z^{10}+192y^{2}z^{12}-7z^{14})}$

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