Properties

Label 2.64.a_abp
Base Field $\F_{2^{6}}$
Dimension $2$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
Weil polynomial:  $( 1 - 13 x + 64 x^{2} )( 1 + 13 x + 64 x^{2} )$
Frobenius angles:  $\pm0.198106042756$, $\pm0.801893957244$
Angle rank:  $1$ (numerical)
Jacobians:  426

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 426 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4056 16451136 68719911624 281693525577984 1152921502463163576 4722426253610370317376 19342813113840728124788136 79228157538525417810440193024 324518553658426719376279169811864 1329227990841918495641742802133107776

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 65 4015 262145 16790239 1073741825 68720346511 4398046511105 281474959033279 18014398509481985 1152921500319480175

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The isogeny class factors as 1.64.an $\times$ 1.64.n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{6}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.abp 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-87}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{12}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.aba_ll$2$(not in LMFDB)
2.64.ba_ll$2$(not in LMFDB)
2.64.a_bp$4$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.aba_ll$2$(not in LMFDB)
2.64.ba_ll$2$(not in LMFDB)
2.64.a_bp$4$(not in LMFDB)
2.64.an_eb$6$(not in LMFDB)
2.64.n_eb$6$(not in LMFDB)