Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 2 x - 3 x^{2} + 16 x^{3} - 12 x^{4} - 32 x^{5} + 64 x^{6}$ |
| Frobenius angles: | $\pm0.119844691140$, $\pm0.314955599331$, $\pm0.934800290470$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 6.0.399424.1 |
| Galois group: | $D_{6}$ |
| Jacobians: | $4$ |
This isogeny class is simple and geometrically simple, not primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $32$ | $2176$ | $366752$ | $16720384$ | $1116666272$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $7$ | $87$ | $255$ | $1063$ | $4015$ | $16439$ | $66111$ | $263751$ | $1053007$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 1 is hyperelliptic):
- $x^3 y+x^3 z+x y^3+x z^3+y^4=0$
- $x^4+a^2 x^2 y^2+a^2 x^2 y z+a x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $x^4+a x^2 y^2+a x^2 y z+a^2 x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $y^2+(a x^3+a x^2) y=(a+1) x^7+(a+1) x+a+1$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| The endomorphism algebra of this simple isogeny class is 6.0.399424.1. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 3.2.a_ab_ac |
| $\F_{2}$ | 3.2.a_ab_c |
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.4.c_ad_aq | $2$ | 3.16.ak_bx_ahc |
| 3.4.ae_n_abc | $4$ | (not in LMFDB) |
| 3.4.e_n_bc | $4$ | (not in LMFDB) |