Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $256$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{4}\cdot16^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 16C2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.2.27 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&10\\12&5\end{bmatrix}$, $\begin{bmatrix}3&15\\8&5\end{bmatrix}$, $\begin{bmatrix}11&7\\12&5\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^3.C_2^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.2.n.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $4$ |
| Cyclic 16-torsion field degree: | $16$ |
| Full 16-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}$ |
| Newforms: | 256.2.a.a$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} + 2 x y - y^{2} - 2 z w $ |
| $=$ | $2 x w + x t + y z + 2 y w$ | |
| $=$ | $x z - 2 x w + 2 y w - y t$ | |
| $=$ | $z^{2} - 8 w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 6 x^{5} z - x^{4} y^{2} + 9 x^{4} z^{2} + 2 x^{3} y^{2} z - 4 x^{3} z^{3} - 9 x^{2} z^{4} + \cdots - z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 5x^{4} - 5x^{2} + 1 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(-1:1:0)$, $(0:-1:1)$, $(0:1:1)$, $(1:1:0)$ | $(1:1:0)$, $(0:1:1)$, $(0:-1:1)$, $(1:-1:0)$ | $(-1:0:-1:-1/2:1)$, $(0:-1:-1:1/2:1)$, $(0:1:-1:1/2:1)$, $(1:0:-1:-1/2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\,\frac{3072xy^{7}+17408xy^{5}w^{2}+18176xy^{5}wt-46528xy^{5}t^{2}-53760xy^{3}w^{2}t^{2}-118752xy^{3}wt^{3}+5088xy^{3}t^{4}-45904xyw^{2}t^{4}-9132xywt^{5}+1183xyt^{6}-1280y^{8}-1024y^{6}w^{2}-3072y^{6}wt+22208y^{6}t^{2}-79680y^{4}w^{2}t^{2}+34272y^{4}wt^{3}-14152y^{4}t^{4}+28304y^{2}w^{2}t^{4}-18552y^{2}wt^{5}-1183y^{2}t^{6}-1183zwt^{6}+8190w^{2}t^{6}-1024t^{8}}{t(80xy^{5}w-164xy^{5}t-504xy^{3}w^{2}t-12xy^{3}wt^{2}-80xy^{3}t^{3}-168xyw^{2}t^{3}-68xywt^{4}-xyt^{5}-32y^{6}w+84y^{6}t-128y^{4}w^{2}t-120y^{4}wt^{2}+60y^{4}t^{3}-120y^{2}w^{2}t^{3}-12y^{2}wt^{4}+y^{2}t^{5}+zwt^{5})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.2.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}-X^{4}Y^{2}+6X^{5}Z+2X^{3}Y^{2}Z+9X^{4}Z^{2}-4X^{3}Z^{3}+2XY^{2}Z^{3}-9X^{2}Z^{4}+Y^{2}Z^{4}+6XZ^{5}-Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 16.48.2.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x^{3}-2x^{2}y+xy^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x^{8}t+2x^{7}yt-6x^{6}y^{2}t-10x^{5}y^{3}t-10x^{3}y^{5}t+6x^{2}y^{6}t+2xy^{7}t-y^{8}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x^{2}y-2xy^{2}+y^{3}$ |
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.x.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.x.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | 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.dn.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.do.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.dp.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.dq.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.192.3-48.kh.1.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 48.192.3-48.ki.1.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.192.3-48.kj.1.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 48.192.3-48.kk.1.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.288.10-48.cj.1.1 | $48$ | $3$ | $3$ | $10$ | $8$ | $1^{4}\cdot2^{2}$ |
| 48.384.11-48.bn.1.4 | $48$ | $4$ | $4$ | $11$ | $4$ | $1^{5}\cdot2^{2}$ |
| 80.192.3-80.mj.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.mk.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.ml.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.mm.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.18-80.bj.1.4 | $80$ | $5$ | $5$ | $18$ | $?$ | not computed |
| 112.192.3-112.jz.1.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ka.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.kb.1.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.kc.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.jz.1.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.ka.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.kb.1.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.kc.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.mj.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.mk.1.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.ml.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.mm.1.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bkb.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bkc.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bkd.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bke.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.mj.1.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.mk.1.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.ml.1.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.mm.1.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.jz.1.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.ka.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.kb.1.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.kc.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |