Properties

Label 2.83.abi_rm
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:  $( 1 - 18 x + 83 x^{2} )( 1 - 16 x + 83 x^{2} )$
  $1 - 34 x + 454 x^{2} - 2822 x^{3} + 6889 x^{4}$
Frobenius angles:  $\pm0.0496118990883$, $\pm0.158801688027$
Angle rank:  $2$ (numerical)
Jacobians:  $6$
Isomorphism classes:  16

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4488$ $45777600$ $326105714088$ $2252030863104000$ $15516083908976100648$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $50$ $6642$ $570326$ $47452814$ $3939051490$ $326940825474$ $27136055965222$ $2252292264606046$ $186940255313618258$ $15516041184959062482$

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.as $\times$ 1.83.aq 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.ac_aes$2$(not in LMFDB)
2.83.c_aes$2$(not in LMFDB)
2.83.bi_rm$2$(not in LMFDB)