Properties

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

Learn more about

Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $( 1 - 8 x )^{2}( 1 - 8 x + 64 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.333333333333$
Angle rank:  $0$ (numerical)
Jacobians:  0

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 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
$A(\F_{q^r})$ 2793 16515009 68718952449 281406257229825 1152815955785449473 4722294425687923097601 19342785443735548410724353 79228157791897854723881959425 324518553658426690754359001612289 1329227994546975833618426784307478529

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 41 4033 262145 16773121 1073643521 68718428161 4398040219649 281474959933441 18014398509481985 1152921503533105153

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The isogeny class factors as 1.64.aq $\times$ 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 $\F_{2^{36}}$ is 1.68719476736.abdvoy 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 $\F_{2^{36}}$.
Remainder of endomorphism lattice by field

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.ai_a$2$(not in LMFDB)
2.64.i_a$2$(not in LMFDB)
2.64.y_jw$2$(not in LMFDB)
2.64.a_aey$3$(not in LMFDB)
2.64.a_cm$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.ai_a$2$(not in LMFDB)
2.64.i_a$2$(not in LMFDB)
2.64.y_jw$2$(not in LMFDB)
2.64.a_aey$3$(not in LMFDB)
2.64.a_cm$3$(not in LMFDB)
2.64.ai_ey$4$(not in LMFDB)
2.64.i_ey$4$(not in LMFDB)
2.64.abg_ou$6$(not in LMFDB)
2.64.aq_hk$6$(not in LMFDB)
2.64.ai_a$6$(not in LMFDB)
2.64.q_hk$6$(not in LMFDB)
2.64.bg_ou$6$(not in LMFDB)
2.64.aq_ey$12$(not in LMFDB)
2.64.a_acm$12$(not in LMFDB)
2.64.a_ey$12$(not in LMFDB)
2.64.q_ey$12$(not in LMFDB)
2.64.a_a$24$(not in LMFDB)
2.64.ai_cm$30$(not in LMFDB)
2.64.i_cm$30$(not in LMFDB)