Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot8$ | Cusp orbits | $1^{2}$ | ||
| 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: | yes $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 8A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.24.1.22 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&13\\6&1\end{bmatrix}$, $\begin{bmatrix}9&3\\6&7\end{bmatrix}$, $\begin{bmatrix}13&9\\10&7\end{bmatrix}$, $\begin{bmatrix}15&1\\10&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.1.b.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $8$ |
| Cyclic 16-torsion field degree: | $64$ |
| Full 16-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 4x $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(0:0:1)$ | ||
| 256.a1 | $-8$ | $8000$ | $= 2^{6} \cdot 5^{3}$ | $8.987$ | $(2:-4:1)$, $(2:4:1)$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{7x^{2}z^{2}-5xy^{2}z+y^{4}+z^{4}}{z^{2}x^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.1-8.c.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.g.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.i.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.l.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.2-16.c.1.3 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2-16.c.1.5 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2-16.d.1.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2-16.d.1.6 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2-16.e.1.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2-16.e.1.7 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2-16.f.1.2 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 16.48.2-16.f.1.5 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.1-24.s.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.t.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.w.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.x.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.2-48.a.1.2 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2-48.a.1.16 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2-48.b.1.2 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2-48.b.1.16 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2-48.c.1.2 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2-48.c.1.16 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2-48.d.1.2 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2-48.d.1.16 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.72.3-24.b.1.10 | $48$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3-24.b.1.5 | $48$ | $4$ | $4$ | $3$ | $0$ | $1^{2}$ |
| 80.48.1-40.s.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.t.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.w.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.x.1.11 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.2-80.a.1.4 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2-80.a.1.14 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2-80.b.1.8 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2-80.b.1.10 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2-80.c.1.4 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2-80.c.1.14 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2-80.d.1.8 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2-80.d.1.10 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.120.5-40.b.1.8 | $80$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 80.144.5-40.b.1.5 | $80$ | $6$ | $6$ | $5$ | $?$ | not computed |
| 80.240.9-40.dd.1.16 | $80$ | $10$ | $10$ | $9$ | $?$ | not computed |
| 112.48.1-56.s.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.t.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.w.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.x.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.2-112.a.1.4 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2-112.a.1.14 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2-112.b.1.8 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2-112.b.1.10 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2-112.c.1.4 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2-112.c.1.14 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2-112.d.1.8 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2-112.d.1.10 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.192.7-56.b.1.1 | $112$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 112.504.19-56.b.1.15 | $112$ | $21$ | $21$ | $19$ | $?$ | not computed |
| 176.48.1-88.s.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.t.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.w.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.x.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.2-176.a.1.3 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2-176.a.1.13 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2-176.b.1.7 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2-176.b.1.9 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2-176.c.1.3 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2-176.c.1.13 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2-176.d.1.7 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2-176.d.1.9 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.288.11-88.b.1.15 | $176$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 208.48.1-104.s.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.t.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.w.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.x.1.11 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.2-208.a.1.4 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2-208.a.1.14 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2-208.b.1.8 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2-208.b.1.10 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2-208.c.1.4 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2-208.c.1.14 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2-208.d.1.8 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2-208.d.1.10 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.336.13-104.b.1.3 | $208$ | $14$ | $14$ | $13$ | $?$ | not computed |
| 240.48.1-120.s.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.t.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.w.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.x.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.2-240.a.1.2 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2-240.a.1.32 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2-240.b.1.2 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2-240.b.1.32 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2-240.c.1.2 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2-240.c.1.32 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2-240.d.1.2 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2-240.d.1.32 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.1-136.s.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.t.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.w.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.x.1.11 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.2-272.a.1.9 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2-272.a.1.13 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2-272.b.1.13 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2-272.b.1.15 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2-272.c.1.13 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2-272.c.1.15 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2-272.d.1.9 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2-272.d.1.11 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.432.17-136.b.1.14 | $272$ | $18$ | $18$ | $17$ | $?$ | not computed |
| 304.48.1-152.s.1.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.t.1.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.w.1.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.x.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.2-304.a.1.4 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2-304.a.1.14 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2-304.b.1.8 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2-304.b.1.10 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2-304.c.1.4 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2-304.c.1.14 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2-304.d.1.8 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2-304.d.1.10 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.480.19-152.b.1.16 | $304$ | $20$ | $20$ | $19$ | $?$ | not computed |