Properties

Label 2.64.aba_ll
Base field $\F_{2^{6}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $( 1 - 13 x + 64 x^{2} )^{2}$
  $1 - 26 x + 297 x^{2} - 1664 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.198106042756$, $\pm0.198106042756$
Angle rank:  $1$ (numerical)
Jacobians:  $21$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2704$ $16451136$ $68876853136$ $281693525577984$ $1153062186256958224$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $39$ $4015$ $262743$ $16790239$ $1073872839$ $68720346511$ $4398049433271$ $281474959033279$ $18014398092657447$ $1152921500319480175$

Jacobians and polarizations

This isogeny class contains the Jacobians of 21 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{6}}$
The isogeny class factors as 1.64.an 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-87}) \)$)$

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.a_abp$2$(not in LMFDB)
2.64.ba_ll$2$(not in LMFDB)
2.64.n_eb$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.64.a_abp$2$(not in LMFDB)
2.64.ba_ll$2$(not in LMFDB)
2.64.n_eb$3$(not in LMFDB)
2.64.a_bp$4$(not in LMFDB)
2.64.an_eb$6$(not in LMFDB)