Invariants
| Level: | $16$ | $\SL_2$-level: | $8$ | 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 | $8^{6}$ | 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: | 8A2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.2.4 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}15&2\\2&1\end{bmatrix}$, $\begin{bmatrix}15&2\\6&1\end{bmatrix}$, $\begin{bmatrix}15&14\\0&11\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.D_8$ |
| Contains $-I$: | no $\quad$ (see 16.48.2.a.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $8$ |
| Cyclic 16-torsion field degree: | $32$ |
| 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 w + y t - z w + z t $ |
| $=$ | $ - x t + y w - z w - z t$ | |
| $=$ | $x^{2} + y^{2} - 2 z^{2}$ | |
| $=$ | $x y + 2 w^{2} + 2 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 2 x^{4} y^{2} + 3 x^{4} z^{2} - 12 x^{2} y^{2} z^{2} + 3 x^{2} z^{4} + 2 y^{2} z^{4} + 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 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(-1:-1:1)$, $(1:1:1)$, $(1:-1:1)$, $(-1:1:1)$ | $(1:-1:0)$, $(0:-1:1)$, $(0:1:1)$, $(1:1:0)$ | $(-2:2:-2:-1:1)$, $(-2:2:2:1:1)$, $(2:-2:-2:1:1)$, $(2:-2:2:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{192xz^{3}t^{4}-5760xzt^{6}+192yz^{3}t^{4}-5760yzt^{6}-4z^{8}+288z^{4}t^{4}-8448z^{2}t^{6}+37w^{8}+900w^{6}t^{2}+8318w^{4}t^{4}+13956w^{2}t^{6}+6565t^{8}}{(w^{2}+t^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.2.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}+2X^{4}Y^{2}+3X^{4}Z^{2}-12X^{2}Y^{2}Z^{2}+3X^{2}Z^{4}+2Y^{2}Z^{4}+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.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}w^{3}+\frac{1}{2}w^{2}t-\frac{1}{2}wt^{2}+\frac{1}{2}t^{3}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{1}{4}zw^{8}+zw^{6}t^{2}+\frac{5}{2}zw^{4}t^{4}+zw^{2}t^{6}-\frac{1}{4}zt^{8}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -\frac{1}{2}w^{3}-\frac{1}{2}w^{2}t-\frac{1}{2}wt^{2}-\frac{1}{2}t^{3}$ |
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.a.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.a.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.b.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.f.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.h.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.j.1.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.192.3-48.i.1.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.192.3-48.k.1.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.192.3-48.r.1.2 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 48.192.3-48.t.1.4 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 48.288.10-48.a.1.11 | $48$ | $3$ | $3$ | $10$ | $8$ | $1^{4}\cdot2^{2}$ |
| 48.384.11-48.a.1.21 | $48$ | $4$ | $4$ | $11$ | $4$ | $1^{5}\cdot2^{2}$ |
| 80.192.3-80.i.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.k.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.r.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.t.1.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.18-80.a.1.8 | $80$ | $5$ | $5$ | $18$ | $?$ | not computed |
| 112.192.3-112.i.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.k.1.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.r.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.t.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.i.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.k.1.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.r.1.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.t.1.4 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.i.1.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.k.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.r.1.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.t.1.4 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bc.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.be.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bt.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bv.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.i.1.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.k.1.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.r.1.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.t.1.3 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.i.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.k.1.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.r.1.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.t.1.4 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |