Properties

Label 2.64.ax_jg
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:  $( 1 - 8 x )^{2}( 1 - 7 x + 64 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.355864001265$
Angle rank:  $1$ (numerical)
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2842 16574544 68712946666 281370287671200 1152797017209387802 4722297626273133409776 19342794148380484204114762 79228161442857269538792532800 324518552709952644592019441491066 1329227992855497229005498750000265104

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 42 4048 262122 16770976 1073625882 68718474736 4398042198858 281474972904256 18014398456831098 1152921502065981328

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The isogeny class factors as 1.64.aq $\times$ 1.64.ah 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.aj_q$2$(not in LMFDB)
2.64.j_q$2$(not in LMFDB)
2.64.x_jg$2$(not in LMFDB)
2.64.b_cu$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.aj_q$2$(not in LMFDB)
2.64.j_q$2$(not in LMFDB)
2.64.x_jg$2$(not in LMFDB)
2.64.b_cu$3$(not in LMFDB)
2.64.ah_ey$4$(not in LMFDB)
2.64.h_ey$4$(not in LMFDB)
2.64.ap_hc$6$(not in LMFDB)
2.64.ab_cu$6$(not in LMFDB)
2.64.p_hc$6$(not in LMFDB)