Properties

Label 2.243.ack_cdr
Base field $\F_{3^{5}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3^{5}}$
Dimension:  $2$
L-polynomial:  $( 1 - 31 x + 243 x^{2} )^{2}$
  $1 - 62 x + 1447 x^{2} - 15066 x^{3} + 59049 x^{4}$
Frobenius angles:  $\pm0.0339262533067$, $\pm0.0339262533067$
Angle rank:  $1$ (numerical)

This isogeny class is not simple, not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $45369$ $3431030625$ $205684817824656$ $12156915630659765625$ $717895300637162526680169$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $182$ $58100$ $14334524$ $3486569348$ $847285438082$ $205891086040550$ $50031544441963574$ $12157665449879954948$ $2954312706426007289732$ $717897987690212974740500$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{5}}$.

Endomorphism algebra over $\F_{3^{5}}$
The isogeny class factors as 1.243.abf 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{5}}$.

SubfieldPrimitive Model
$\F_{3}$2.3.ac_h

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.243.a_ash$2$(not in LMFDB)
2.243.ck_cdr$2$(not in LMFDB)
2.243.bf_bbq$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.243.a_ash$2$(not in LMFDB)
2.243.ck_cdr$2$(not in LMFDB)
2.243.bf_bbq$3$(not in LMFDB)
2.243.a_sh$4$(not in LMFDB)
2.243.abf_bbq$6$(not in LMFDB)