Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $128$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $8^{4}\cdot16^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16C5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.192.5.265 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&4\\12&7\end{bmatrix}$, $\begin{bmatrix}7&14\\8&13\end{bmatrix}$, $\begin{bmatrix}13&2\\8&3\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_4:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.96.5.p.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $4$ |
| Cyclic 16-torsion field degree: | $16$ |
| Full 16-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{33}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 32.2.a.a, 128.2.a.a, 128.2.a.c, 128.2.b.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 y^{2} + 2 y z + t^{2} $ |
| $=$ | $2 y z - 2 z^{2} + w^{2}$ | |
| $=$ | $4 x^{2} - w t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{8} - 3 x^{4} y^{2} z^{2} + y^{4} z^{4} + y^{2} z^{6} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:-1/2:1/2:1:0)$, $(0:1/2:-1/2:1:0)$ |
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 16.48.3.d.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-Y^{3}Z+YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 16.96.5.p.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{8}-3X^{4}Y^{2}Z^{2}+Y^{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 |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.96.1-8.i.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.96.3-16.d.2.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3-16.d.2.9 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3-16.f.2.8 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 16.96.3-16.f.2.12 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.384.9-16.ba.2.6 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 16.384.9-16.bc.2.1 | $16$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 16.384.9-16.bq.4.2 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 16.384.9-16.bw.1.6 | $16$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.384.9-48.dw.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.dz.2.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.384.9-48.fp.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.fs.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.576.21-48.me.1.20 | $48$ | $3$ | $3$ | $21$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.768.25-48.gm.2.17 | $48$ | $4$ | $4$ | $25$ | $2$ | $1^{10}\cdot2^{5}$ |
| 80.384.9-80.gu.2.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.gx.1.6 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.iv.1.15 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.jc.2.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.dw.1.7 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.dz.2.2 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fp.2.6 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fs.1.5 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.dw.1.7 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.dz.2.2 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.fp.2.6 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.fs.1.5 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gu.2.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gx.1.6 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.iv.1.15 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.jc.2.2 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.xy.1.14 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.yh.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bdr.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bea.1.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gu.2.1 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gx.1.6 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.iv.1.15 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.jc.2.6 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.dw.1.7 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.dz.2.2 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.fp.2.6 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.fs.1.5 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |