Properties

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

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $( 1 - 15 x + 64 x^{2} )( 1 - 8 x + 64 x^{2} )$
Frobenius angles:  $\pm0.113134082257$, $\pm0.333333333333$
Angle rank:  $1$ (numerical)
Jacobians:  6

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2850 16644000 68858168850 281523306888000 1152900736926299250 4722349644870882444000 19342818534837223169380050 79228176264475395472476432000 324518563321925232852664256646450 1329227999290944231826880336654100000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 42 4064 262674 16780096 1073722482 68719231712 4398047743698 281475025561216 18014399045913906 1152921507647841824

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The isogeny class factors as 1.64.ap $\times$ 1.64.ai 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 $\F_{2^{18}}$ is 1.262144.atb $\times$ 1.262144.bnk. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{18}}$.

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_i$2$(not in LMFDB)
2.64.h_i$2$(not in LMFDB)
2.64.x_jo$2$(not in LMFDB)
2.64.b_aei$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.ah_i$2$(not in LMFDB)
2.64.h_i$2$(not in LMFDB)
2.64.x_jo$2$(not in LMFDB)
2.64.b_aei$3$(not in LMFDB)
2.64.abf_oe$6$(not in LMFDB)
2.64.ab_aei$6$(not in LMFDB)
2.64.bf_oe$6$(not in LMFDB)
2.64.ap_ey$12$(not in LMFDB)
2.64.p_ey$12$(not in LMFDB)