Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.223 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&6\\0&5\end{bmatrix}$, $\begin{bmatrix}11&7\\8&7\end{bmatrix}$, $\begin{bmatrix}11&9\\8&9\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8.C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.j.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| Full 16-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} - 2 x y + z^{2} $ |
| $=$ | $8 x^{2} + 14 x y - 2 y^{2} - z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 x^{2} y^{2} - 4 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 5\,\frac{4608y^{2}z^{8}w^{2}+26496y^{2}z^{4}w^{6}+1638y^{2}w^{10}+1280z^{12}-13824z^{10}w^{2}+55536z^{8}w^{4}-79488z^{6}w^{6}+27717z^{4}w^{8}-4914z^{2}w^{10}+461w^{12}}{w^{2}z^{4}(32y^{2}z^{4}+2y^{2}w^{4}-96z^{6}-41z^{4}w^{2}-6z^{2}w^{4}-w^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.1.j.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{8}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{2}Y^{2}-4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.r.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.h.1.13 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.h.1.16 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.r.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.b.1.13 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.b.1.16 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.192.1-16.r.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.r.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.s.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.s.2.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 32.192.5-32.k.1.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 32.192.5-32.l.1.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.192.5-32.l.2.2 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.192.5-32.m.1.8 | $32$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 48.192.1-48.cb.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cb.2.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cc.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cc.2.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.bn.1.16 | $48$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
| 48.384.9-48.mr.1.16 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{8}$ |
| 80.192.1-80.cb.1.6 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cb.2.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cc.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cc.2.11 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.t.1.16 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 96.192.5-96.k.1.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.l.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.l.2.13 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.m.1.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.1-112.cb.1.6 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cb.2.7 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cc.1.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cc.2.7 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 160.192.5-160.k.1.15 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.l.1.13 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.l.2.7 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.m.1.15 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.1-176.cb.1.6 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cb.2.7 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cc.1.4 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cc.2.7 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cb.1.6 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cb.2.6 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cc.1.4 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cc.2.11 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 224.192.5-224.k.1.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.l.1.13 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.l.2.7 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.m.1.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.gf.1.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gf.2.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gg.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gg.2.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cb.1.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cb.2.8 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cc.1.8 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cc.2.4 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cb.1.6 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cb.2.6 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cc.1.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cc.2.7 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |