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Properties

Label 16.48.1.q.1
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: X346
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.64

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&3\\0&13\end{bmatrix}$, $\begin{bmatrix}1&3\\8&1\end{bmatrix}$, $\begin{bmatrix}11&5\\8&5\end{bmatrix}$, $\begin{bmatrix}13&14\\0&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.q.1.1, 16.96.1-16.q.1.2, 16.96.1-16.q.1.3, 16.96.1-16.q.1.4, 16.96.1-16.q.1.5, 16.96.1-16.q.1.6, 16.96.1-16.q.1.7, 16.96.1-16.q.1.8, 48.96.1-16.q.1.1, 48.96.1-16.q.1.2, 48.96.1-16.q.1.3, 48.96.1-16.q.1.4, 48.96.1-16.q.1.5, 48.96.1-16.q.1.6, 48.96.1-16.q.1.7, 48.96.1-16.q.1.8, 80.96.1-16.q.1.1, 80.96.1-16.q.1.2, 80.96.1-16.q.1.3, 80.96.1-16.q.1.4, 80.96.1-16.q.1.5, 80.96.1-16.q.1.6, 80.96.1-16.q.1.7, 80.96.1-16.q.1.8, 112.96.1-16.q.1.1, 112.96.1-16.q.1.2, 112.96.1-16.q.1.3, 112.96.1-16.q.1.4, 112.96.1-16.q.1.5, 112.96.1-16.q.1.6, 112.96.1-16.q.1.7, 112.96.1-16.q.1.8, 176.96.1-16.q.1.1, 176.96.1-16.q.1.2, 176.96.1-16.q.1.3, 176.96.1-16.q.1.4, 176.96.1-16.q.1.5, 176.96.1-16.q.1.6, 176.96.1-16.q.1.7, 176.96.1-16.q.1.8, 208.96.1-16.q.1.1, 208.96.1-16.q.1.2, 208.96.1-16.q.1.3, 208.96.1-16.q.1.4, 208.96.1-16.q.1.5, 208.96.1-16.q.1.6, 208.96.1-16.q.1.7, 208.96.1-16.q.1.8, 240.96.1-16.q.1.1, 240.96.1-16.q.1.2, 240.96.1-16.q.1.3, 240.96.1-16.q.1.4, 240.96.1-16.q.1.5, 240.96.1-16.q.1.6, 240.96.1-16.q.1.7, 240.96.1-16.q.1.8, 272.96.1-16.q.1.1, 272.96.1-16.q.1.2, 272.96.1-16.q.1.3, 272.96.1-16.q.1.4, 272.96.1-16.q.1.5, 272.96.1-16.q.1.6, 272.96.1-16.q.1.7, 272.96.1-16.q.1.8, 304.96.1-16.q.1.1, 304.96.1-16.q.1.2, 304.96.1-16.q.1.3, 304.96.1-16.q.1.4, 304.96.1-16.q.1.5, 304.96.1-16.q.1.6, 304.96.1-16.q.1.7, 304.96.1-16.q.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

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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2}\cdot\frac{1392x^{2}y^{14}-700162336x^{2}y^{12}z^{2}-405727951872x^{2}y^{10}z^{4}+919304715623424x^{2}y^{8}z^{6}+15432878312521728x^{2}y^{6}z^{8}-420356841293957627904x^{2}y^{4}z^{10}-143534890256669545070592x^{2}y^{2}z^{12}-13434757656044767260180480x^{2}z^{14}-649264xy^{14}z-2765800704xy^{12}z^{3}+212710031616xy^{10}z^{5}+8925826873995264xy^{8}z^{7}-1942776008963129344xy^{6}z^{9}-4172430243473724014592xy^{4}z^{11}-1200895072816697680330752xy^{2}z^{13}-102867981249586667464949760xz^{15}-y^{16}+102966144y^{14}z^{2}-73888864512y^{12}z^{4}+69404275863552y^{10}z^{6}+33879480217059328y^{8}z^{8}-37547147115692556288y^{6}z^{10}-21527670089259069997056y^{4}z^{12}-3659542046852944094035968y^{2}z^{14}-196515802501630393713688576z^{16}}{y^{2}(x^{2}y^{12}+143008x^{2}y^{10}z^{2}+948649088x^{2}y^{8}z^{4}+1376540606464x^{2}y^{6}z^{6}+636136197632000x^{2}y^{4}z^{8}+88820748315656192x^{2}y^{2}z^{10}+262144x^{2}z^{12}+96xy^{12}z+3408608xy^{10}z^{3}+13386764800xy^{8}z^{5}+14474026002432xy^{6}z^{7}+5544287377326080xy^{4}z^{9}+680087524195500032xy^{2}z^{11}-1048576xz^{13}+4544y^{12}z^{2}+60143104y^{10}z^{4}+130729266944y^{8}z^{6}+85486924120064y^{6}z^{8}+19618585931087872y^{4}z^{10}+1299218123733336064y^{2}z^{12}-7340032z^{14})}$

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