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 $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 16C2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.2.135 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&12\\4&5\end{bmatrix}$, $\begin{bmatrix}5&13\\12&11\end{bmatrix}$, $\begin{bmatrix}7&14\\8&7\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8:\OD_{16}$ |
| Contains $-I$: | no $\quad$ (see 16.48.2.o.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: | $2^{16}$ |
| 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} + y^{2} + 2 z w $ |
| $=$ | $ - 4 x w + x t + y z + y t$ | |
| $=$ | $x z - x t + 4 y w + y t$ | |
| $=$ | $z^{2} + 16 w^{2} - 2 t^{2}$ |
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.a1 | $-16$ | $287496$ | $= 2^{3} \cdot 3^{3} \cdot 11^{3}$ | $12.569$ | $(1:-1:1)$, $(-1:1:1)$, $(-1:-1:1)$, $(1:1:1)$ | $(0:1:1)$, $(1:1:0)$, $(1:-1:0)$, $(0:-1:1)$ | $(-1/2:-1/2:-1:1/4:1)$, $(-1/2:1/2:1:-1/4:1)$, $(1/2:-1/2:1:-1/4:1)$, $(1/2:1/2:-1:1/4:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^3\,\frac{4096xy^{5}w^{2}-11264xy^{5}wt-14720xy^{5}t^{2}-355200xy^{3}w^{2}t^{2}-107616xy^{3}wt^{3}+307584xy^{3}t^{4}-358400xyw^{2}t^{4}+734508xywt^{5}-95167xyt^{6}+256y^{8}+11776y^{6}wt+3456y^{6}t^{2}+57600y^{4}w^{2}t^{2}-73056y^{4}wt^{3}-66472y^{4}t^{4}-1056960y^{2}w^{2}t^{4}+206028y^{2}wt^{5}+95167y^{2}t^{6}+95167zwt^{6}+524160w^{2}t^{6}-65536t^{8}}{t(128xy^{5}w+160xy^{5}t-384xy^{3}w^{2}t+96xy^{3}wt^{2}+40xy^{3}t^{3}-384xyw^{2}t^{3}+76xywt^{4}-xyt^{5}-128y^{6}w-32y^{6}t-640y^{4}w^{2}t-288y^{4}wt^{2}-80y^{4}t^{3}+96y^{2}w^{2}t^{3}+124y^{2}wt^{4}+y^{2}t^{5}+zwt^{5})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.2.o.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
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.o.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}x^{3}+\frac{1}{2}x^{2}y-\frac{1}{2}xy^{2}+\frac{1}{2}y^{3}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{1}{4}x^{8}t+x^{6}y^{2}t+\frac{5}{2}x^{4}y^{4}t+x^{2}y^{6}t-\frac{1}{4}y^{8}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -\frac{1}{2}x^{3}-\frac{1}{2}x^{2}y-\frac{1}{2}xy^{2}-\frac{1}{2}y^{3}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.0-8.y.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.y.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ | 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.dr.1.2 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.ds.1.3 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.dv.1.3 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 16.192.3-16.dw.1.3 | $16$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.192.3-48.kl.1.6 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 48.192.3-48.km.1.6 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 48.192.3-48.kn.1.7 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.192.3-48.ko.1.6 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.288.10-48.ck.1.21 | $48$ | $3$ | $3$ | $10$ | $8$ | $1^{4}\cdot2^{2}$ |
| 48.384.11-48.bo.1.11 | $48$ | $4$ | $4$ | $11$ | $4$ | $1^{5}\cdot2^{2}$ |
| 80.192.3-80.mn.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.mo.1.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.mp.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.mq.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.18-80.bk.1.10 | $80$ | $5$ | $5$ | $18$ | $?$ | not computed |
| 112.192.3-112.kd.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ke.1.7 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.kf.1.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.kg.1.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.kd.1.6 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.ke.1.6 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.kf.1.6 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.3-176.kg.1.6 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.mn.1.6 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.mo.1.7 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.mp.1.6 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.192.3-208.mq.1.6 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bkf.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bkg.1.15 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bkh.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bki.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.mn.1.8 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.mo.1.5 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.mp.1.4 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.192.3-272.mq.1.6 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.kd.1.6 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.ke.1.6 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.kf.1.7 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.192.3-304.kg.1.6 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |