Properties

Label 2.64.az_km
Base Field $\F_{2^{6}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $( 1 - 8 x )^{2}( 1 - 9 x + 64 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.309839631512$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2744 16447536 68712424424 281437900380000 1152840305690007704 4722297901148943205776 19342778756208740794944584 79228152438505363164788280000 324518552630200417591949193187064 1329227995667562387786429070824101616

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 40 4016 262120 16775008 1073666200 68718478736 4398038699080 281474940914368 18014398452403960 1152921504505059056

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The isogeny class factors as 1.64.aq $\times$ 1.64.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{6}}$.

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.ah_aq$2$(not in LMFDB)
2.64.h_aq$2$(not in LMFDB)
2.64.z_km$2$(not in LMFDB)
2.64.ab_ce$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.ah_aq$2$(not in LMFDB)
2.64.h_aq$2$(not in LMFDB)
2.64.z_km$2$(not in LMFDB)
2.64.ab_ce$3$(not in LMFDB)
2.64.aj_ey$4$(not in LMFDB)
2.64.j_ey$4$(not in LMFDB)
2.64.ar_hs$6$(not in LMFDB)
2.64.b_ce$6$(not in LMFDB)
2.64.r_hs$6$(not in LMFDB)