Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $4$ are rational) | Cusp widths | $4^{16}\cdot16^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.384.5.369 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&8\\0&9\end{bmatrix}$, $\begin{bmatrix}5&12\\0&1\end{bmatrix}$, $\begin{bmatrix}15&13\\0&9\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2:C_2^2$ |
| Contains $-I$: | no $\quad$ (see 16.192.5.bu.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $1$ |
| Cyclic 16-torsion field degree: | $4$ |
| Full 16-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{28}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 32.2.a.a$^{2}$, 64.2.a.a, 64.2.b.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - y z $ |
| $=$ | $y^{2} - z^{2} + t^{2}$ | |
| $=$ | $y^{2} + z^{2} - 4 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} - 32 x^{6} z^{2} - 40 x^{4} z^{4} - 8 x^{2} z^{6} - y^{4} z^{4} + z^{8} $ |
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.
| Canonical model |
|---|
| $(0:0:1:1/2:1)$, $(0:0:-1:1/2:1)$, $(0:0:-1:-1/2:1)$, $(0:0:1:-1/2:1)$ |
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.cn.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2w-t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w-t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2Y^{3}Z+2YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 16.192.5.bu.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y+z+2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{8}-32X^{6}Z^{2}-40X^{4}Z^{4}-Y^{4}Z^{4}-8X^{2}Z^{6}+Z^{8} $ |
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.l.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.192.1-8.l.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.192.1-16.l.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.192.1-16.l.2.5 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.192.1-16.m.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.192.1-16.m.1.7 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.192.3-16.ck.1.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.192.3-16.ck.1.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.192.3-16.cn.1.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.cn.1.8 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.192.3-16.cp.1.3 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.192.3-16.cp.1.6 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.192.3-16.cs.2.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.192.3-16.cs.2.4 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.bi.1.2 | $16$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 16.768.13-16.bj.1.1 | $16$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 32.768.13-32.bd.2.3 | $32$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 32.768.13-32.bi.1.6 | $32$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 32.768.13-32.by.1.2 | $32$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 32.768.13-32.cd.2.4 | $32$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 32.768.17-32.ca.1.2 | $32$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{3}$ |
| 32.768.17-32.ca.2.3 | $32$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{3}$ |
| 48.768.13-48.ji.1.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.768.13-48.jj.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.1152.37-48.fcl.2.2 | $48$ | $3$ | $3$ | $37$ | $3$ | $1^{16}\cdot2^{6}\cdot4$ |
| 48.1536.41-48.bqx.2.3 | $48$ | $4$ | $4$ | $41$ | $1$ | $1^{18}\cdot2^{7}\cdot4$ |