Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-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, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.97 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&0\\0&13\end{bmatrix}$, $\begin{bmatrix}3&3\\8&3\end{bmatrix}$, $\begin{bmatrix}9&14\\0&5\end{bmatrix}$, $\begin{bmatrix}11&11\\8&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $\OD_{32}.C_2^3$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.v.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $4$ |
| Full 16-torsion field degree: | $256$ |
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 $ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.bb.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.1.15 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.bb.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.b.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.b.1.16 | $16$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.192.1-16.f.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.i.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.m.2.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.s.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.3-16.dz.1.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.ea.2.4 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.eb.2.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.ec.1.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.z.1.3 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.ba.1.2 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.bb.1.4 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.bc.2.7 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.5-32.s.1.15 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.192.5-32.w.1.15 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 32.192.5-32.ba.1.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.192.5-32.be.1.4 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.1-48.co.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cs.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.de.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.di.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.3-48.kt.1.8 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.ku.2.16 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.kv.2.8 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.kw.1.4 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.288.9-48.ez.1.7 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bba.1.7 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.cn.2.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cr.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.dd.2.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.dh.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.3-80.nl.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.nm.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.nn.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.no.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.17-80.cf.1.7 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 96.192.3-96.bh.1.3 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bi.1.3 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bj.1.11 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bk.2.19 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.5-96.bu.1.27 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.by.1.19 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cs.2.30 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cw.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.1-112.cn.2.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cr.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.dd.2.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.dh.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.3-112.kl.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.km.2.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.kn.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ko.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.bp.1.5 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.bq.1.3 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.br.1.7 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.bs.1.13 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.5-160.bu.1.29 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.by.1.25 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cs.2.16 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cw.1.8 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.1-176.cn.2.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cr.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.dd.2.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.dh.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.3-176.kl.1.3 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.km.2.5 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.kn.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.ko.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.1-208.cn.2.6 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cr.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.dd.2.6 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.dh.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.3-208.nl.1.3 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.nm.2.5 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.nn.1.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.no.1.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.bh.1.2 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.bi.1.2 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.bj.1.4 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.bk.2.4 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.5-224.bu.1.29 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.by.1.25 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cs.2.16 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cw.1.8 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.jm.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ju.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ks.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.la.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-240.bld.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ble.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.blf.2.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.blg.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.1-272.cn.2.6 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cr.2.10 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.dd.2.3 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.dh.2.6 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.3-272.nl.1.11 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.nm.1.7 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.nn.1.7 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.no.2.13 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.1-304.cn.2.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cr.1.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.dd.2.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.dh.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.3-304.kl.1.3 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.km.2.5 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.kn.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.ko.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |