Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $4^{4}\cdot16^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16C2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.2.9 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&8\\8&1\end{bmatrix}$, $\begin{bmatrix}9&6\\0&13\end{bmatrix}$, $\begin{bmatrix}15&4\\8&1\end{bmatrix}$, $\begin{bmatrix}15&6\\0&9\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.2.d.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $16$ |
| Full 16-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{12}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $2$ |
| Newforms: | 64.2.b.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x z w + y w^{2} $ |
| $=$ | $2 x z^{2} + y z w$ | |
| $=$ | $2 x y z + y^{2} w$ | |
| $=$ | $2 x^{2} z + x y w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} - x^{3} z^{2} - x^{2} y^{2} z - y^{2} z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{5} + x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:1:0:0)$, $(0:0:-1/2:1)$, $(0:0:0:1)$, $(0:0:1/2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{60x^{2}y^{6}w^{2}+66x^{2}y^{2}w^{6}-xy^{9}+48xy^{5}w^{4}-129xyw^{8}+124y^{8}z^{2}+98y^{8}w^{2}+132y^{6}zw^{3}+184y^{4}z^{2}w^{4}+2y^{4}w^{6}-120y^{2}zw^{7}-124z^{2}w^{8}+32w^{10}}{w^{4}y^{4}(xy+4z^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}-X^{2}Y^{2}Z-X^{3}Z^{2}-Y^{2}Z^{3} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 16.48.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}zw$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}yz^{4}w+\frac{1}{16}yz^{2}w^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z^{2}$ |
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.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.i.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.192.3-16.o.1.6 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 16.192.3-16.t.2.3 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 16.192.3-16.v.2.6 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 16.192.3-16.w.2.4 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 16.192.3-16.x.2.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 16.192.3-16.y.2.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 16.192.3-16.z.2.6 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 16.192.3-16.ba.1.8 | $16$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 48.192.3-48.bn.1.11 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.192.3-48.bo.2.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 48.192.3-48.bp.2.10 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.192.3-48.bq.2.10 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.192.3-48.bs.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 48.192.3-48.bt.1.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.192.3-48.bu.2.9 | $48$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 48.192.3-48.bv.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 48.288.10-48.j.2.20 | $48$ | $3$ | $3$ | $10$ | $0$ | $1^{4}\cdot4$ |
| 48.384.11-48.g.1.18 | $48$ | $4$ | $4$ | $11$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 80.192.3-80.cd.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.ce.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.cf.2.10 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.cg.1.12 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.ci.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.cj.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.ck.2.11 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.cl.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.18-80.f.1.20 | $80$ | $5$ | $5$ | $18$ | $?$ | not computed |
| 112.192.3-112.bn.1.11 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bo.2.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bp.2.10 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bq.2.11 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bs.1.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bt.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bu.2.9 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bv.1.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bn.1.11 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bo.2.6 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bp.2.10 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bq.2.11 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bs.2.5 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bt.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bu.2.9 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.bv.1.9 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.cd.1.3 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.ce.2.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.cf.2.10 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.cg.1.12 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.ci.1.6 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.cj.1.3 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.ck.2.11 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.cl.1.3 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.dz.1.23 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ea.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.eb.2.26 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ec.2.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ee.2.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ef.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.eg.2.25 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.eh.1.19 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.cd.2.5 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.ce.2.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.cf.2.13 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.cg.1.17 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.ci.1.12 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.cj.1.7 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.ck.2.10 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.cl.2.3 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bn.1.11 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bo.2.6 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bp.2.10 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bq.2.11 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bs.1.5 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bt.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bu.2.9 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.bv.1.5 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |