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: | 16D5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.192.5.267 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&14\\4&9\end{bmatrix}$, $\begin{bmatrix}11&0\\0&7\end{bmatrix}$, $\begin{bmatrix}11&0\\8&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_4:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.96.5.q.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.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y t + z w $ |
| $=$ | $2 x^{2} - z w$ | |
| $=$ | $8 y^{2} + 2 z^{2} - 2 w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} y^{2} - x^{4} z^{2} + 4 y^{4} z^{2} - 4 y^{2} 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 |
|---|
| $(0:1/2:0:1:0)$, $(0:-1/2:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{256z^{12}-384z^{10}t^{2}+912z^{8}t^{4}-752z^{6}t^{6}+858z^{4}t^{8}-339z^{2}t^{10}+256w^{12}+384w^{10}t^{2}+912w^{8}t^{4}+752w^{6}t^{6}+858w^{4}t^{8}+339w^{2}t^{10}+256t^{12}}{t^{4}(16z^{8}-16z^{6}t^{2}+2z^{4}t^{4}+z^{2}t^{6}+16w^{8}+16w^{6}t^{2}+2w^{4}t^{4}-w^{2}t^{6})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 16.96.5.q.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}Y^{2}-X^{4}Z^{2}+4Y^{4}Z^{2}-4Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.96.1-8.i.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 16.96.3-16.e.1.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3-16.e.1.9 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3-16.f.1.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 16.96.3-16.f.1.10 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
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.1.6 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 16.384.9-16.bc.2.1 | $16$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 16.384.9-16.bq.3.2 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 16.384.9-16.bw.1.6 | $16$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 32.384.13-32.u.2.7 | $32$ | $2$ | $2$ | $13$ | $3$ | $2^{2}\cdot4$ |
| 32.384.13-32.u.4.7 | $32$ | $2$ | $2$ | $13$ | $3$ | $2^{2}\cdot4$ |
| 32.384.13-32.v.1.5 | $32$ | $2$ | $2$ | $13$ | $3$ | $2^{2}\cdot4$ |
| 32.384.13-32.v.3.5 | $32$ | $2$ | $2$ | $13$ | $3$ | $2^{2}\cdot4$ |
| 48.384.9-48.dx.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.ea.2.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.384.9-48.fq.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.ft.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.576.21-48.ml.2.6 | $48$ | $3$ | $3$ | $21$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.25-48.gr.1.9 | $48$ | $4$ | $4$ | $25$ | $2$ | $1^{10}\cdot2^{3}\cdot4$ |
| 80.384.9-80.gv.2.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.gy.1.3 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.iw.1.15 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.jd.1.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.13-96.bu.3.2 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 96.384.13-96.bu.4.2 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 96.384.13-96.bw.1.9 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 96.384.13-96.bw.3.5 | $96$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 112.384.9-112.dx.1.7 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.ea.2.2 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fq.2.6 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.ft.1.5 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 160.384.13-160.bu.2.11 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 160.384.13-160.bu.4.11 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 160.384.13-160.bw.1.9 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 160.384.13-160.bw.3.9 | $160$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 176.384.9-176.dx.1.7 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.ea.2.2 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.fq.2.6 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.ft.1.5 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gv.1.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gy.1.3 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.iw.1.15 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.jd.1.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 224.384.13-224.bu.3.2 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 224.384.13-224.bu.4.2 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 224.384.13-224.bw.1.13 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 224.384.13-224.bw.3.13 | $224$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.384.9-240.xz.1.15 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.yi.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bds.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.beb.1.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gv.2.1 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gy.1.5 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.iw.1.15 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.jd.2.3 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.dx.1.7 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.ea.2.2 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.fq.2.6 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.ft.1.5 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |