Properties

Label 2.64.abg_ou
Base Field $\F_{2^{6}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $( 1 - 8 x )^{4}$
Frobenius angles:  $0$, $0$, $0$, $0$
Angle rank:  $0$ (numerical)
Jacobians:  1

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2401 15752961 68184176641 281200199450625 1152780773560811521 4722294425687923097601 19342776220372307646873601 79228143624800094964756250625 324518543987020277952513142947841 1329227990833155722679814984029962241

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 33 3841 260097 16760833 1073610753 68718428161 4398038122497 281474909601793 18014397972611073 1152921500311879681

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The isogeny class factors as 1.64.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{6}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.a_ae
$\F_{2}$2.2.a_c

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.a_aey$2$(not in LMFDB)
2.64.bg_ou$2$(not in LMFDB)
2.64.ai_a$3$(not in LMFDB)
2.64.q_hk$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.a_aey$2$(not in LMFDB)
2.64.bg_ou$2$(not in LMFDB)
2.64.ai_a$3$(not in LMFDB)
2.64.q_hk$3$(not in LMFDB)
2.64.aq_ey$4$(not in LMFDB)
2.64.a_ey$4$(not in LMFDB)
2.64.q_ey$4$(not in LMFDB)
2.64.i_cm$5$(not in LMFDB)
2.64.ay_jw$6$(not in LMFDB)
2.64.aq_hk$6$(not in LMFDB)
2.64.a_cm$6$(not in LMFDB)
2.64.i_a$6$(not in LMFDB)
2.64.y_jw$6$(not in LMFDB)
2.64.a_a$8$(not in LMFDB)
2.64.ai_cm$10$(not in LMFDB)
2.64.ai_ey$12$(not in LMFDB)
2.64.a_acm$12$(not in LMFDB)
2.64.i_ey$12$(not in LMFDB)