Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| 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 $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.24.1.25 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}7&3\\2&13\end{bmatrix}$, $\begin{bmatrix}7&12\\12&11\end{bmatrix}$, $\begin{bmatrix}11&10\\10&13\end{bmatrix}$, $\begin{bmatrix}13&7\\6&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.1.c.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: | $2^{6}$ |
| 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 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 | $\infty$ | $0.000$ | $(0:1:0)$, $(0:0:1)$ | ||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(-2:0:1)$, $(2:0:1)$ |
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.b.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.f.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.n.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.o.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.q.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.r.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.s.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.t.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.y.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.z.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.ba.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-8.bb.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.y.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.z.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bc.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bd.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bg.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bh.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bi.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bj.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bo.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bp.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.bq.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-24.br.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.72.3-24.c.1.5 | $48$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3-24.c.1.9 | $48$ | $4$ | $4$ | $3$ | $0$ | $1^{2}$ |
| 80.48.1-40.y.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.z.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bc.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bd.1.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bg.1.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bh.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bi.1.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bj.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bo.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bp.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.bq.1.6 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-40.br.1.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.120.5-40.c.1.6 | $80$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 80.144.5-40.c.1.31 | $80$ | $6$ | $6$ | $5$ | $?$ | not computed |
| 80.240.9-40.de.1.25 | $80$ | $10$ | $10$ | $9$ | $?$ | not computed |
| 112.48.1-56.y.1.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.z.1.6 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bc.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bd.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bg.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bh.1.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bi.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bj.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bo.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bp.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.bq.1.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.1-56.br.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.7-56.c.1.13 | $112$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 112.504.19-56.c.1.25 | $112$ | $21$ | $21$ | $19$ | $?$ | not computed |
| 176.48.1-88.y.1.4 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.z.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bc.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bd.1.6 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bg.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bh.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bi.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bj.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bo.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bp.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.bq.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.1-88.br.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.288.11-88.c.1.8 | $176$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 208.48.1-104.y.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.z.1.8 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bc.1.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bd.1.8 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bg.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bh.1.6 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bi.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bj.1.6 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bo.1.8 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bp.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.bq.1.8 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.1-104.br.1.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.336.13-104.c.1.29 | $208$ | $14$ | $14$ | $13$ | $?$ | not computed |
| 240.48.1-120.y.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.z.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bc.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bd.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bg.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bh.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bi.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bj.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bo.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bp.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.bq.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-120.br.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.y.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.z.1.4 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bc.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bd.1.4 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bg.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bh.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bi.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bj.1.3 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bo.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bp.1.4 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.bq.1.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.1-136.br.1.4 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.432.17-136.c.1.10 | $272$ | $18$ | $18$ | $17$ | $?$ | not computed |
| 304.48.1-152.y.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.z.1.6 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bc.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bd.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bg.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bh.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bi.1.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bj.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bo.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bp.1.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.bq.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.1-152.br.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.480.19-152.c.1.6 | $304$ | $20$ | $20$ | $19$ | $?$ | not computed |