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Properties

Label 16.48.1.v.1
Level $16$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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

Other labels

Cummins and Pauli (CP) label: 16G1
Rouse and Zureick-Brown (RZB) label: X340
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.45

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}3&15\\8&11\end{bmatrix}$, $\begin{bmatrix}7&0\\0&15\end{bmatrix}$, $\begin{bmatrix}9&2\\0&1\end{bmatrix}$, $\begin{bmatrix}11&9\\8&7\end{bmatrix}$, $\begin{bmatrix}13&3\\8&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.v.1.1, 16.96.1-16.v.1.2, 16.96.1-16.v.1.3, 16.96.1-16.v.1.4, 16.96.1-16.v.1.5, 16.96.1-16.v.1.6, 16.96.1-16.v.1.7, 16.96.1-16.v.1.8, 16.96.1-16.v.1.9, 16.96.1-16.v.1.10, 16.96.1-16.v.1.11, 16.96.1-16.v.1.12, 32.96.1-16.v.1.1, 32.96.1-16.v.1.2, 32.96.1-16.v.1.3, 32.96.1-16.v.1.4, 32.96.1-16.v.1.5, 32.96.1-16.v.1.6, 32.96.1-16.v.1.7, 32.96.1-16.v.1.8, 48.96.1-16.v.1.1, 48.96.1-16.v.1.2, 48.96.1-16.v.1.3, 48.96.1-16.v.1.4, 48.96.1-16.v.1.5, 48.96.1-16.v.1.6, 48.96.1-16.v.1.7, 48.96.1-16.v.1.8, 48.96.1-16.v.1.9, 48.96.1-16.v.1.10, 48.96.1-16.v.1.11, 48.96.1-16.v.1.12, 80.96.1-16.v.1.1, 80.96.1-16.v.1.2, 80.96.1-16.v.1.3, 80.96.1-16.v.1.4, 80.96.1-16.v.1.5, 80.96.1-16.v.1.6, 80.96.1-16.v.1.7, 80.96.1-16.v.1.8, 80.96.1-16.v.1.9, 80.96.1-16.v.1.10, 80.96.1-16.v.1.11, 80.96.1-16.v.1.12, 96.96.1-16.v.1.1, 96.96.1-16.v.1.2, 96.96.1-16.v.1.3, 96.96.1-16.v.1.4, 96.96.1-16.v.1.5, 96.96.1-16.v.1.6, 96.96.1-16.v.1.7, 96.96.1-16.v.1.8, 112.96.1-16.v.1.1, 112.96.1-16.v.1.2, 112.96.1-16.v.1.3, 112.96.1-16.v.1.4, 112.96.1-16.v.1.5, 112.96.1-16.v.1.6, 112.96.1-16.v.1.7, 112.96.1-16.v.1.8, 112.96.1-16.v.1.9, 112.96.1-16.v.1.10, 112.96.1-16.v.1.11, 112.96.1-16.v.1.12, 160.96.1-16.v.1.1, 160.96.1-16.v.1.2, 160.96.1-16.v.1.3, 160.96.1-16.v.1.4, 160.96.1-16.v.1.5, 160.96.1-16.v.1.6, 160.96.1-16.v.1.7, 160.96.1-16.v.1.8, 176.96.1-16.v.1.1, 176.96.1-16.v.1.2, 176.96.1-16.v.1.3, 176.96.1-16.v.1.4, 176.96.1-16.v.1.5, 176.96.1-16.v.1.6, 176.96.1-16.v.1.7, 176.96.1-16.v.1.8, 176.96.1-16.v.1.9, 176.96.1-16.v.1.10, 176.96.1-16.v.1.11, 176.96.1-16.v.1.12, 208.96.1-16.v.1.1, 208.96.1-16.v.1.2, 208.96.1-16.v.1.3, 208.96.1-16.v.1.4, 208.96.1-16.v.1.5, 208.96.1-16.v.1.6, 208.96.1-16.v.1.7, 208.96.1-16.v.1.8, 208.96.1-16.v.1.9, 208.96.1-16.v.1.10, 208.96.1-16.v.1.11, 208.96.1-16.v.1.12, 224.96.1-16.v.1.1, 224.96.1-16.v.1.2, 224.96.1-16.v.1.3, 224.96.1-16.v.1.4, 224.96.1-16.v.1.5, 224.96.1-16.v.1.6, 224.96.1-16.v.1.7, 224.96.1-16.v.1.8, 240.96.1-16.v.1.1, 240.96.1-16.v.1.2, 240.96.1-16.v.1.3, 240.96.1-16.v.1.4, 240.96.1-16.v.1.5, 240.96.1-16.v.1.6, 240.96.1-16.v.1.7, 240.96.1-16.v.1.8, 240.96.1-16.v.1.9, 240.96.1-16.v.1.10, 240.96.1-16.v.1.11, 240.96.1-16.v.1.12, 272.96.1-16.v.1.1, 272.96.1-16.v.1.2, 272.96.1-16.v.1.3, 272.96.1-16.v.1.4, 272.96.1-16.v.1.5, 272.96.1-16.v.1.6, 272.96.1-16.v.1.7, 272.96.1-16.v.1.8, 272.96.1-16.v.1.9, 272.96.1-16.v.1.10, 272.96.1-16.v.1.11, 272.96.1-16.v.1.12, 304.96.1-16.v.1.1, 304.96.1-16.v.1.2, 304.96.1-16.v.1.3, 304.96.1-16.v.1.4, 304.96.1-16.v.1.5, 304.96.1-16.v.1.6, 304.96.1-16.v.1.7, 304.96.1-16.v.1.8, 304.96.1-16.v.1.9, 304.96.1-16.v.1.10, 304.96.1-16.v.1.11, 304.96.1-16.v.1.12
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 $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(2:0:1)$, $(1:-2:1)$, $(1:2:1)$

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

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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.e.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.i.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.m.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.1.s.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.96.3.dz.1 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.ea.2 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.eb.2 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.ec.1 $16$ $2$ $2$ $3$ $0$ $2$
32.96.3.z.1 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.ba.1 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.bb.1 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.bc.2 $32$ $2$ $2$ $3$ $0$ $2$
32.96.5.s.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.96.5.w.1 $32$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
32.96.5.ba.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.96.5.be.1 $32$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.1.co.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cs.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.de.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.di.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.3.kt.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.ku.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.kv.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.kw.1 $48$ $2$ $2$ $3$ $0$ $2$
48.144.9.ez.1 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bba.1 $48$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
80.96.1.cn.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.cr.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.dd.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.1.dh.2 $80$ $2$ $2$ $1$ $?$ dimension zero
80.96.3.nl.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.nm.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.nn.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.no.1 $80$ $2$ $2$ $3$ $?$ not computed
80.240.17.cf.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.fx.1 $80$ $6$ $6$ $17$ $?$ not computed
96.96.3.bh.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.bi.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.bj.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.bk.2 $96$ $2$ $2$ $3$ $?$ not computed
96.96.5.bu.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.by.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cs.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cw.1 $96$ $2$ $2$ $5$ $?$ not computed
112.96.1.cn.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.cr.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.dd.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.1.dh.2 $112$ $2$ $2$ $1$ $?$ dimension zero
112.96.3.kl.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.km.2 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.kn.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.ko.1 $112$ $2$ $2$ $3$ $?$ not computed
160.96.3.bp.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.bq.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.br.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.bs.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.5.bu.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.by.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.cs.2 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.cw.1 $160$ $2$ $2$ $5$ $?$ not computed
176.96.1.cn.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.cr.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.dd.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.1.dh.2 $176$ $2$ $2$ $1$ $?$ dimension zero
176.96.3.kl.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.km.2 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.kn.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.ko.1 $176$ $2$ $2$ $3$ $?$ not computed
208.96.1.cn.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.cr.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.dd.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.1.dh.2 $208$ $2$ $2$ $1$ $?$ dimension zero
208.96.3.nl.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.nm.2 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.nn.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.no.1 $208$ $2$ $2$ $3$ $?$ not computed
224.96.3.bh.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.bi.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.bj.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.bk.2 $224$ $2$ $2$ $3$ $?$ not computed
224.96.5.bu.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.by.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.cs.2 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.cw.1 $224$ $2$ $2$ $5$ $?$ not computed
240.96.1.jm.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ju.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ks.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.la.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.bld.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ble.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.blf.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.blg.1 $240$ $2$ $2$ $3$ $?$ not computed
272.96.1.cn.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.cr.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.dd.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.1.dh.2 $272$ $2$ $2$ $1$ $?$ dimension zero
272.96.3.nl.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.nm.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.nn.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.no.2 $272$ $2$ $2$ $3$ $?$ not computed
304.96.1.cn.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.cr.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.dd.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.1.dh.2 $304$ $2$ $2$ $1$ $?$ dimension zero
304.96.3.kl.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.km.2 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.kn.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.ko.1 $304$ $2$ $2$ $3$ $?$ not computed