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 $2$ 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, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.151 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&3\\8&11\end{bmatrix}$, $\begin{bmatrix}5&1\\0&3\end{bmatrix}$, $\begin{bmatrix}5&3\\0&15\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8.C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.x.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| 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 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{24x^{2}y^{14}+21586x^{2}y^{12}z^{2}+3441624x^{2}y^{10}z^{4}+193245609x^{2}y^{8}z^{6}+4400033856x^{2}y^{6}z^{8}+38490501129x^{2}y^{4}z^{10}+142771224564x^{2}y^{2}z^{12}+190919389185x^{2}z^{14}+316xy^{14}z+146472xy^{12}z^{3}+17836719xy^{10}z^{5}+861766122xy^{8}z^{7}+17968009096xy^{6}z^{9}+151627161576xy^{4}z^{11}+552379830297xy^{2}z^{13}+730920968190xz^{15}+y^{16}+2832y^{14}z^{2}+676068y^{12}z^{4}+51635064y^{10}z^{6}+1673064092y^{8}z^{8}+24815025696y^{6}z^{10}+173617528794y^{4}z^{12}+566431350864y^{2}z^{14}+698164379641z^{16}}{z^{5}y^{2}(307x^{2}y^{6}z+112528x^{2}y^{4}z^{3}+8659504x^{2}y^{2}z^{5}+175629440x^{2}z^{7}+xy^{8}+2854xy^{6}z^{2}+637168xy^{4}z^{4}+38479136xy^{2}z^{6}+672384512xz^{8}+24y^{8}z+18528y^{6}z^{3}+2153152y^{4}z^{5}+72453504y^{2}z^{7}+642251264z^{9})}$ |
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.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.1.14 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.bb.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.b.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.b.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.j.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.o.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.s.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 32.192.5-32.u.1.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.192.5-32.y.1.8 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 32.192.5-32.bc.1.3 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.192.5-32.bg.1.3 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.1-48.cq.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cu.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dg.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dk.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.fn.1.17 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bbg.1.10 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.cp.1.5 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.ct.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.df.1.5 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.dj.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.cl.1.9 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 96.192.5-96.bw.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.ca.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cu.1.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cy.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.1-112.cp.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.ct.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.df.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.dj.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 160.192.5-160.bw.1.14 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.ca.2.11 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cu.1.12 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cy.1.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.1-176.cp.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.ct.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.df.2.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.dj.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cp.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.ct.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.df.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.dj.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 224.192.5-224.bw.1.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.ca.2.13 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cu.1.12 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cy.1.10 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.js.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ka.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ky.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.lg.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cp.1.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.ct.1.3 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.df.2.3 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.dj.2.3 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cp.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.ct.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.df.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.dj.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |