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: | 16E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.11 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&6\\0&13\end{bmatrix}$, $\begin{bmatrix}11&2\\8&5\end{bmatrix}$, $\begin{bmatrix}15&8\\0&1\end{bmatrix}$, $\begin{bmatrix}15&10\\8&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.b.2 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} - x $ |
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 |
|---|
| $(0:0:1)$, $(-1:0:1)$, $(0:1:0)$, $(1:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{25x^{2}y^{12}z^{2}+51x^{2}y^{8}z^{6}+15x^{2}y^{4}z^{10}+8xy^{14}z+69xy^{10}z^{5}+44xy^{6}z^{9}+6xy^{2}z^{13}+y^{16}+38y^{12}z^{4}+32y^{8}z^{8}+6y^{4}z^{12}+z^{16}}{z^{5}y^{4}(5x^{2}y^{4}z+4x^{2}z^{5}+xy^{6}+12xy^{2}z^{4}+8y^{4}z^{3})}$ |
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.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.i.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.i.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.i.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.d.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.d.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.e.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.e.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.3-16.s.2.4 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.192.3-16.x.1.6 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.x.2.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.ba.1.8 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.ba.2.3 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.bb.1.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.1-48.k.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.k.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.3-48.ci.1.4 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.192.3-48.cm.1.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cm.2.9 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.co.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.co.2.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cp.1.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.288.9-48.h.2.3 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 48.384.9-48.hs.2.6 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{8}$ |
| 80.192.1-80.k.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.k.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.3-80.dk.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.do.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.do.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.dq.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.dq.2.15 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.dr.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.17-80.d.2.6 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 112.192.1-112.k.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.k.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.3-112.ci.2.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cm.1.9 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cm.2.9 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.co.1.9 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.co.2.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cp.1.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.1-176.k.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.k.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.3-176.ci.2.6 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.cm.1.9 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.cm.2.5 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.co.1.9 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.co.2.9 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.cp.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.1-208.k.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.k.2.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.3-208.dk.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.do.1.5 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.do.2.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.dq.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.dq.2.15 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.dr.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.1-240.w.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.w.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-240.iz.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jj.1.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jj.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jl.1.19 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jl.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jn.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.1-272.k.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.k.2.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.3-272.dk.1.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.do.1.11 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.do.2.9 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.dq.1.17 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.dq.2.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.dr.1.4 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.1-304.k.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.k.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.3-304.ci.2.6 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.cm.1.5 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.cm.2.9 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.co.1.5 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.co.2.9 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.cp.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |