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Properties

Label 16.48.1.q.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: $64$
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: 16G1
Rouse and Zureick-Brown (RZB) label: X345
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.57

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&10\\8&3\end{bmatrix}$, $\begin{bmatrix}3&1\\0&7\end{bmatrix}$, $\begin{bmatrix}7&0\\0&7\end{bmatrix}$, $\begin{bmatrix}7&11\\8&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.q.2.1, 16.96.1-16.q.2.2, 16.96.1-16.q.2.3, 16.96.1-16.q.2.4, 16.96.1-16.q.2.5, 16.96.1-16.q.2.6, 16.96.1-16.q.2.7, 16.96.1-16.q.2.8, 48.96.1-16.q.2.1, 48.96.1-16.q.2.2, 48.96.1-16.q.2.3, 48.96.1-16.q.2.4, 48.96.1-16.q.2.5, 48.96.1-16.q.2.6, 48.96.1-16.q.2.7, 48.96.1-16.q.2.8, 80.96.1-16.q.2.1, 80.96.1-16.q.2.2, 80.96.1-16.q.2.3, 80.96.1-16.q.2.4, 80.96.1-16.q.2.5, 80.96.1-16.q.2.6, 80.96.1-16.q.2.7, 80.96.1-16.q.2.8, 112.96.1-16.q.2.1, 112.96.1-16.q.2.2, 112.96.1-16.q.2.3, 112.96.1-16.q.2.4, 112.96.1-16.q.2.5, 112.96.1-16.q.2.6, 112.96.1-16.q.2.7, 112.96.1-16.q.2.8, 176.96.1-16.q.2.1, 176.96.1-16.q.2.2, 176.96.1-16.q.2.3, 176.96.1-16.q.2.4, 176.96.1-16.q.2.5, 176.96.1-16.q.2.6, 176.96.1-16.q.2.7, 176.96.1-16.q.2.8, 208.96.1-16.q.2.1, 208.96.1-16.q.2.2, 208.96.1-16.q.2.3, 208.96.1-16.q.2.4, 208.96.1-16.q.2.5, 208.96.1-16.q.2.6, 208.96.1-16.q.2.7, 208.96.1-16.q.2.8, 240.96.1-16.q.2.1, 240.96.1-16.q.2.2, 240.96.1-16.q.2.3, 240.96.1-16.q.2.4, 240.96.1-16.q.2.5, 240.96.1-16.q.2.6, 240.96.1-16.q.2.7, 240.96.1-16.q.2.8, 272.96.1-16.q.2.1, 272.96.1-16.q.2.2, 272.96.1-16.q.2.3, 272.96.1-16.q.2.4, 272.96.1-16.q.2.5, 272.96.1-16.q.2.6, 272.96.1-16.q.2.7, 272.96.1-16.q.2.8, 304.96.1-16.q.2.1, 304.96.1-16.q.2.2, 304.96.1-16.q.2.3, 304.96.1-16.q.2.4, 304.96.1-16.q.2.5, 304.96.1-16.q.2.6, 304.96.1-16.q.2.7, 304.96.1-16.q.2.8
Cyclic 16-isogeny field degree: $2$
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} - 44x - 112 $
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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
$(-4:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^2}\cdot\frac{48x^{2}y^{14}-368864x^{2}y^{12}z^{2}+1564511232x^{2}y^{10}z^{4}-118879404186624x^{2}y^{8}z^{6}-35827354485915648x^{2}y^{6}z^{8}+5486059308791169024x^{2}y^{4}z^{10}-236513474548762411008x^{2}y^{2}z^{12}+3279970130870308700160x^{2}z^{14}-1616xy^{14}z+74037504xy^{12}z^{3}-391934689536xy^{10}z^{5}-1242500991135744xy^{8}z^{7}-230282299356348416xy^{6}z^{9}+40179742035806257152xy^{4}z^{11}-1786078758714467155968xy^{2}z^{13}+25114253234762353213440xz^{15}+y^{16}-115584y^{14}z^{2}+1541236992y^{12}z^{4}-9120470863872y^{10}z^{6}-6474529098907648y^{8}z^{8}+136296699068940288y^{6}z^{10}+52469341391619096576y^{4}z^{12}-3078722489027691282432y^{2}z^{14}+47977490845124490428416z^{16}}{zy^{2}(1340x^{2}y^{10}z+6493696x^{2}y^{8}z^{3}+8644488448x^{2}y^{6}z^{5}+4671313813504x^{2}y^{4}z^{7}+1101563548418048x^{2}y^{2}z^{9}+94290337626849280x^{2}z^{11}+xy^{12}+28336xy^{10}z^{2}+87808768xy^{8}z^{4}+92761120768xy^{6}z^{6}+43175964020736xy^{4}z^{8}+9149470046355456xy^{2}z^{10}+721967372344229888xz^{12}+48y^{12}z+443744y^{10}z^{3}+822085632y^{8}z^{5}+582369900544y^{6}z^{7}+187200444301312y^{4}z^{9}+27061695578439680y^{2}z^{11}+1379224087347331072z^{13})}$

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.ba.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.24.0.e.1 $16$ $2$ $2$ $0$ $0$ 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.a.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.g.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.l.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.p.1 $16$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cf.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cj.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cv.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cz.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.ei.2 $48$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
48.192.9.bar.1 $48$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
80.96.1.ce.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.ci.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.cu.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.cy.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.240.17.bw.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.fk.1 $80$ $6$ $6$ $17$ $?$ not computed
112.96.1.ce.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.ci.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.cu.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.cy.2 $112$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.ce.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.ci.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.cu.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.cy.2 $176$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.ce.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.ci.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cu.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cy.2 $208$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.iv.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.jd.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.kb.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.kj.2 $240$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.ce.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.ci.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cu.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cy.2 $272$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.ce.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.ci.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cu.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cy.2 $304$ $2$ $2$ $1$ $?$ dimension zero