Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $128$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{12}\cdot16^{4}$ | Cusp orbits | $2^{6}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.384.5.28 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&12\\8&1\end{bmatrix}$, $\begin{bmatrix}9&0\\8&13\end{bmatrix}$, $\begin{bmatrix}13&10\\12&15\end{bmatrix}$, $\begin{bmatrix}13&14\\4&3\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2:C_2^2$ |
| Contains $-I$: | no $\quad$ (see 16.192.5.d.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $4$ |
| Full 16-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{33}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1\cdot2^{2}$ |
| Newforms: | 32.2.a.a, 128.2.e.b$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y z + w t $ |
| $=$ | $2 x^{2} - y z - z^{2} + w t - t^{2}$ | |
| $=$ | $2 x^{2} - y^{2} + y z - w^{2} - w t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{4} - 2 x^{4} y^{2} z^{2} + x^{4} z^{4} - 4 x^{2} y^{4} z^{2} - 4 x^{2} y^{2} z^{4} + \cdots + y^{2} z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 16.96.3.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-X^{2}Y^{2}+Y^{3}Z-X^{2}Z^{2}-2Y^{2}Z^{2}-YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 16.192.5.d.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{4}-2X^{4}Y^{2}Z^{2}+X^{4}Z^{4}-4X^{2}Y^{4}Z^{2}-4X^{2}Y^{2}Z^{4}+Y^{6}Z^{2}+2Y^{4}Z^{4}+Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.1-8.g.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 16.192.1-8.g.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 16.192.2-16.b.1.4 | $16$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 16.192.2-16.b.1.12 | $16$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 16.192.2-16.f.2.4 | $16$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 16.192.2-16.f.2.13 | $16$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 16.192.3-16.n.1.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.n.1.15 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.768.13-16.g.1.3 | $16$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 16.768.13-16.i.1.1 | $16$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 16.768.13-16.j.1.3 | $16$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 16.768.13-16.k.1.1 | $16$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 16.768.17-16.e.2.4 | $16$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 16.768.17-16.f.1.2 | $16$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 32.768.17-32.a.2.1 | $32$ | $2$ | $2$ | $17$ | $4$ | $4\cdot8$ |
| 32.768.17-32.b.2.3 | $32$ | $2$ | $2$ | $17$ | $0$ | $4\cdot8$ |
| 32.768.17-32.e.2.3 | $32$ | $2$ | $2$ | $17$ | $0$ | $4\cdot8$ |
| 32.768.17-32.f.1.1 | $32$ | $2$ | $2$ | $17$ | $0$ | $4\cdot8$ |
| 48.768.13-48.q.1.5 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.r.1.3 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.s.2.5 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.t.2.3 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.s.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.t.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.1152.37-48.du.1.3 | $48$ | $3$ | $3$ | $37$ | $1$ | $1^{8}\cdot2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bd.1.11 | $48$ | $4$ | $4$ | $41$ | $0$ | $1^{8}\cdot2^{6}\cdot8^{2}$ |