Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2^{2}\cdot4\cdot16$ | Cusp orbits | $1^{4}$ | ||
| 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: | 16A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.1.79 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&15\\0&11\end{bmatrix}$, $\begin{bmatrix}5&8\\0&11\end{bmatrix}$, $\begin{bmatrix}7&11\\8&1\end{bmatrix}$, $\begin{bmatrix}7&12\\8&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.24.1.a.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| 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} - 4x $ |
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 |
|---|
| $(0:0:1)$, $(-2:0:1)$, $(0:1:0)$, $(2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{10848x^{2}y^{4}z^{2}-4193280x^{2}z^{6}-176xy^{6}z+1037568xy^{2}z^{5}+y^{8}-264960y^{4}z^{4}+16777216z^{8}}{zy^{2}(8x^{2}y^{2}z+xy^{4}+64xz^{4}+16y^{2}z^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.n.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.a.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.d.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.g.1.10 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.i.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.q.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.q.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.r.1.7 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.r.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.s.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.s.2.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.t.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.t.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.q.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.r.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.u.1.10 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.v.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bg.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bg.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bh.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bh.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bi.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bi.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bj.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bj.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.144.5-48.a.1.33 | $48$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 48.192.5-48.mh.1.34 | $48$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 80.96.1-80.q.1.11 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.r.1.5 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.u.1.13 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.v.1.5 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bg.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bg.2.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bh.1.9 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bh.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bi.1.9 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bi.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bj.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1-80.bj.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.240.9-80.a.1.21 | $80$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 80.288.9-80.i.1.43 | $80$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 80.480.17-80.cm.1.41 | $80$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 112.96.1-112.q.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.r.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.u.1.10 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.v.1.6 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bg.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bg.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bh.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bh.2.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bi.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bi.2.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bj.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.1-112.bj.2.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.384.13-112.g.1.2 | $112$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 176.96.1-176.q.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.r.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.u.1.10 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.v.1.10 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bg.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bg.2.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bh.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bh.2.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bi.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bi.2.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bj.1.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.1-176.bj.2.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.q.1.11 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.r.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.u.1.13 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.v.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bg.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bg.2.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bh.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bh.2.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bi.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bi.2.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bj.1.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.1-208.bj.2.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.q.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.r.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.u.1.18 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.v.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bg.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bg.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bh.1.17 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bh.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bi.1.17 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bi.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bj.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bj.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.q.1.13 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.r.1.15 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.u.1.9 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.v.1.9 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bg.1.10 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bg.2.14 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bh.1.14 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bh.2.10 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bi.1.14 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bi.2.10 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bj.1.12 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.1-272.bj.2.10 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.q.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.r.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.u.1.10 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.v.1.6 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bg.1.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bg.2.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bh.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bh.2.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bi.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bi.2.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bj.1.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.1-304.bj.2.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |