Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.192.1.140 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&5\\0&9\end{bmatrix}$, $\begin{bmatrix}9&1\\8&13\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $\OD_{32}:C_4$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.r.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| Full 16-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - y^{2} - z^{2} $ |
| $=$ | $2 x^{2} + y^{2} + 3 z^{2} + w^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{(16z^{8}-224z^{6}w^{2}-40z^{4}w^{4}+8z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{4}(2z^{2}+w^{2})^{8}(4z^{2}+w^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.o.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.l.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.l.2.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-8.o.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.y.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.y.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.ba.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.ba.2.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.j.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.j.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.u.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.u.2.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.w.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.w.1.7 | $16$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 32.384.9-32.bp.1.1 | $32$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 32.384.9-32.br.2.3 | $32$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.qs.2.7 | $48$ | $3$ | $3$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.qu.2.7 | $48$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 96.384.9-96.dz.1.2 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.ed.2.5 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 160.384.9-160.ed.1.2 | $160$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 160.384.9-160.eh.2.5 | $160$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 224.384.9-224.dz.1.2 | $224$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 224.384.9-224.ed.2.5 | $224$ | $2$ | $2$ | $9$ | $?$ | not computed |