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: | $0$ | ||||||
| $\Q$-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: | 16.192.5.268 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&6\\0&3\end{bmatrix}$, $\begin{bmatrix}13&6\\4&5\end{bmatrix}$, $\begin{bmatrix}15&10\\4&3\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_4:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.96.5.m.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.b, 128.2.a.d, 128.2.b.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 y^{2} + w^{2} + w t $ |
| $=$ | $2 z^{2} + w t - t^{2}$ | |
| $=$ | $2 x^{2} - y z$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{8} + 3 x^{4} y^{2} z^{2} - 4 y^{6} z^{2} + y^{4} z^{4} $ |
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 |
|---|
| $(0:0:-1:-1:1)$, $(0:0:1:-1:1)$ |
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 16.48.3.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t$ |
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.m.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{8}+3X^{4}Y^{2}Z^{2}-4Y^{6}Z^{2}+Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.96.1-8.i.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.96.3-16.c.2.8 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.96.3-16.c.2.11 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.96.3-16.d.1.7 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3-16.d.1.11 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.384.9-16.ba.1.6 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 16.384.9-16.bb.2.1 | $16$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 16.384.9-16.bq.3.2 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 16.384.9-16.bt.1.6 | $16$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 32.384.13-32.q.2.7 | $32$ | $2$ | $2$ | $13$ | $0$ | $4^{2}$ |
| 32.384.13-32.q.4.7 | $32$ | $2$ | $2$ | $13$ | $0$ | $4^{2}$ |
| 32.384.13-32.r.1.5 | $32$ | $2$ | $2$ | $13$ | $2$ | $2^{4}$ |
| 32.384.13-32.r.3.5 | $32$ | $2$ | $2$ | $13$ | $2$ | $2^{4}$ |
| 48.384.9-48.dr.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.dt.2.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.fk.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.fm.1.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.576.21-48.ij.1.20 | $48$ | $3$ | $3$ | $21$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.768.25-48.dr.2.17 | $48$ | $4$ | $4$ | $25$ | $2$ | $1^{10}\cdot2^{5}$ |
| 80.384.9-80.gp.2.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.gr.1.6 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.im.1.15 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.is.2.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.13-96.bc.3.2 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 96.384.13-96.bc.4.2 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 96.384.13-96.bd.1.9 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 96.384.13-96.bd.3.5 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 112.384.9-112.dr.1.7 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.dt.2.2 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fk.2.6 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fm.1.5 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 160.384.13-160.bc.2.11 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 160.384.13-160.bc.4.11 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 160.384.13-160.bd.1.9 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 160.384.13-160.bd.3.9 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 176.384.9-176.dr.1.7 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.dt.2.2 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.fk.2.6 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.fm.1.5 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gp.1.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gr.1.6 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.im.1.15 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.is.2.2 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 224.384.13-224.bc.3.2 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 224.384.13-224.bc.4.2 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 224.384.13-224.bd.1.13 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 224.384.13-224.bd.3.13 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.384.9-240.wr.1.15 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.wt.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bck.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bcm.1.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gp.2.1 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gr.1.6 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.im.1.15 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.is.2.6 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.dr.1.7 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.dt.2.2 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.fk.2.6 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.fm.1.5 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |