Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| 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 | $2^{4}\cdot4^{2}$ | ||
| 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: | 16.192.1.107 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&14\\0&11\end{bmatrix}$, $\begin{bmatrix}13&13\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $\OD_{32}:C_4$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.h.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $1$ |
| Cyclic 16-torsion field degree: | $4$ |
| Full 16-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y + y^{2} - z^{2} $ |
| $=$ | $x^{2} + 3 x y + 3 y^{2} + 5 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 x^{2} y^{2} + 6 x^{2} z^{2} + z^{4} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{8}+16z^{6}w^{2}+20z^{4}w^{4}+8z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{16}(2z^{2}+w^{2})^{2}(4z^{2}+w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.96.1.h.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{2}Y^{2}+6X^{2}Z^{2}+Z^{4} $ |
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.m.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.f.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.f.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-8.m.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.w.2.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.w.2.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.x.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.x.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.d.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.d.1.8 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.s.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.s.2.7 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.t.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.t.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 |
|---|---|---|---|---|---|---|
| 32.384.5-32.g.1.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.384.5-32.k.2.1 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.576.17-48.fr.1.1 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.jo.1.1 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 96.384.5-96.q.1.7 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.w.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.u.1.7 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.be.2.2 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.q.1.7 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.w.2.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |