Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}( 1 - 3 x^{2} + 4 x^{4} )$ |
$1 - 7 x + 23 x^{2} - 41 x^{3} + 30 x^{4} + 34 x^{5} - 104 x^{6} + 68 x^{7} + 120 x^{8} - 328 x^{9} + 368 x^{10} - 224 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.115026728081$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.884973271919$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4$ | $4000$ | $2276092$ | $64000000$ | $1640595484$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $2$ | $26$ | $38$ | $46$ | $74$ | $94$ | $254$ | $422$ | $1082$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 3 $\times$ 1.2.ab $\times$ 2.2.a_ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab 3 $\times$ 1.16.i 3 . The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ad 2 $\times$ 1.4.a 3 $\times$ 1.4.d. The endomorphism algebra for each factor is: - 1.4.ad 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
- 1.4.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
- 1.4.d : \(\Q(\sqrt{-7}) \).
Base change
This is a primitive isogeny class.