Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $256$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $3 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $8^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.3.69 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&10\\0&15\end{bmatrix}$, $\begin{bmatrix}3&11\\4&5\end{bmatrix}$, $\begin{bmatrix}15&7\\8&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^3.C_2^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.3.bc.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $4$ |
| Cyclic 16-torsion field degree: | $32$ |
| Full 16-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{22}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 64.2.a.a, 256.2.a.e |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - x w t + y z t $ |
| $=$ | $ - x z w + y z^{2}$ | |
| $=$ | $ - x w^{2} + y z w$ | |
| $=$ | $ - x y w + y^{2} z$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + x^{3} y^{2} - 4 y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{8} + 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.
| Embedded model |
|---|
| $(0:1:0:0:0)$, $(0:0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^9}\cdot\frac{131072xy^{6}-787456xy^{3}wt^{2}+65544xw^{2}t^{4}-385024y^{5}wt+311232y^{2}w^{2}t^{3}+4095z^{2}t^{5}-t^{7}}{twy^{5}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.3.bc.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}+X^{3}Y^{2}-4YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 16.48.3.bc.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z^{3}t-2w^{4}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.n.1.4 | $8$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 16.48.1-8.n.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.192.5-16.bs.1.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 16.192.5-16.bt.1.5 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 16.192.5-16.ch.1.3 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 16.192.5-16.ci.2.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.192.5-48.gb.1.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.192.5-48.gd.1.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.192.5-48.gn.2.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.192.5-48.gp.2.3 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.288.11-48.ei.1.8 | $48$ | $3$ | $3$ | $11$ | $0$ | $1^{4}\cdot4$ |
| 48.384.13-48.da.2.15 | $48$ | $4$ | $4$ | $13$ | $3$ | $1^{4}\cdot2^{3}$ |
| 80.192.5-80.fz.1.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.gb.1.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.gn.1.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.gp.1.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.480.19-80.cm.2.8 | $80$ | $5$ | $5$ | $19$ | $?$ | not computed |
| 112.192.5-112.fv.1.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.fx.1.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.gh.1.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.gj.2.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5-176.fv.1.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5-176.fx.1.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5-176.gh.1.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5-176.gj.2.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5-208.fz.1.2 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5-208.gb.1.4 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5-208.gn.1.4 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5-208.gp.1.4 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.rz.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.sd.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.sx.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.tb.2.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5-272.fz.2.2 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5-272.gb.1.4 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5-272.gn.2.4 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5-272.gp.1.2 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5-304.fv.1.2 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5-304.fx.1.4 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5-304.gh.1.4 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5-304.gj.2.3 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |