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: | 16.96.1.5 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&14\\0&3\end{bmatrix}$, $\begin{bmatrix}5&8\\0&11\end{bmatrix}$, $\begin{bmatrix}9&10\\0&1\end{bmatrix}$, $\begin{bmatrix}13&2\\0&15\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.b.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $1$ |
| Cyclic 16-torsion field degree: | $4$ |
| 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} + 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 |
|---|
| $(0:1:0)$, $(2:-4:1)$, $(0:0:1)$, $(2:4:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{400x^{2}y^{12}z^{2}-52224x^{2}y^{8}z^{6}+983040x^{2}y^{4}z^{10}-32xy^{14}z+17664xy^{10}z^{5}-720896xy^{6}z^{9}+6291456xy^{2}z^{13}+y^{16}-2432y^{12}z^{4}+131072y^{8}z^{8}-1572864y^{4}z^{12}+16777216z^{16}}{z^{5}y^{4}(20x^{2}y^{4}z-1024x^{2}z^{5}-xy^{6}+768xy^{2}z^{4}-128y^{4}z^{3})}$ |
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.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.g.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.g.1.15 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.i.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.b.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.b.1.15 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.d.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.d.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.f.1.10 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.f.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.3-16.s.1.10 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.192.3-16.z.1.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.z.2.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.bb.1.6 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 32.192.3-32.c.1.9 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.c.2.3 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.d.1.10 | $32$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 32.192.3-32.d.2.4 | $32$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 32.192.5-32.c.1.2 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 32.192.5-32.c.2.2 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 32.192.5-32.d.1.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.192.5-32.d.2.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.192.1-48.j.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.j.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.l.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.l.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.3-48.ci.2.10 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.192.3-48.cn.1.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cn.2.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cp.2.6 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.288.9-48.h.1.1 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 48.384.9-48.hs.1.2 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{8}$ |
| 80.192.1-80.j.1.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.j.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.l.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.l.2.9 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.3-80.dk.2.13 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.dp.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.dp.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.dr.2.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.17-80.d.1.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 96.192.3-96.c.1.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.c.2.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.d.1.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.d.2.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.5-96.c.1.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.c.2.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.d.1.21 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.d.2.21 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.1-112.j.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.j.2.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.l.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.l.2.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.3-112.ci.1.11 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cn.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cn.2.3 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cp.2.11 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.c.1.10 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.c.2.6 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.d.1.9 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.d.2.5 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.5-160.c.1.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.c.2.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.d.1.13 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.d.2.13 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.1-176.j.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.j.2.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.l.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.l.2.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.3-176.ci.1.12 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.cn.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.cn.2.3 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.cp.2.12 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.1-208.j.1.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.j.2.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.l.1.9 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.l.2.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.3-208.dk.2.13 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.dp.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.dp.2.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.dr.2.9 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.c.1.14 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.c.2.14 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.d.1.14 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.d.2.14 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.5-224.c.1.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.c.2.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.d.1.23 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.d.2.23 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.v.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.v.2.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.x.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.x.2.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-240.iz.2.29 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jk.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jk.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jn.2.25 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.1-272.j.1.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.j.2.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.l.1.9 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.l.2.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.3-272.dk.2.14 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.dp.1.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.dp.2.3 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.dr.2.10 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.1-304.j.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.j.2.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.l.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.l.2.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.3-304.ci.1.12 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.cn.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.cn.2.3 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.cp.2.12 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |