Properties

Label 2.83.abe_ox
Base field $\F_{83}$
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:  $\F_{83}$
Dimension:  $2$
L-polynomial:  $( 1 - 17 x + 83 x^{2} )( 1 - 13 x + 83 x^{2} )$
  $1 - 30 x + 387 x^{2} - 2490 x^{3} + 6889 x^{4}$
Frobenius angles:  $\pm0.117184483028$, $\pm0.247123549255$
Angle rank:  $2$ (numerical)
Jacobians:  $6$
Isomorphism classes:  10

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4757$ $46604329$ $327146653424$ $2252881645896361$ $15516538108163352557$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $54$ $6764$ $572148$ $47470740$ $3939166794$ $326941116518$ $27136052904798$ $2252292230561764$ $186940255460442444$ $15516041193252547964$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83}$.

Endomorphism algebra over $\F_{83}$
The isogeny class factors as 1.83.ar $\times$ 1.83.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.83.ae_acd$2$(not in LMFDB)
2.83.e_acd$2$(not in LMFDB)
2.83.be_ox$2$(not in LMFDB)