Properties

Label 2.9.aj_bk
Base field $\F_{3^{2}}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x )^{2}( 1 - 3 x + 9 x^{2} )$
  $1 - 9 x + 36 x^{2} - 81 x^{3} + 81 x^{4}$
Frobenius angles:  $0$, $0$, $\pm0.333333333333$
Angle rank:  $0$ (numerical)
Jacobians:  $0$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $28$ $5824$ $529984$ $42515200$ $3443973148$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $73$ $730$ $6481$ $58321$ $528526$ $4776409$ $43040161$ $387420490$ $3486725353$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{12}}$.

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.ag $\times$ 1.9.ad 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 $\overline{\F}_{3^{2}}$
The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
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
2.9.ad_a$2$2.81.aj_a
2.9.d_a$2$2.81.aj_a
2.9.j_bk$2$2.81.aj_a
2.9.a_as$3$2.729.a_acec
2.9.a_j$3$2.729.a_acec
2.9.j_bk$3$2.729.a_acec

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

TwistExtension degreeCommon base change
2.9.ad_a$2$2.81.aj_a
2.9.d_a$2$2.81.aj_a
2.9.j_bk$2$2.81.aj_a
2.9.a_as$3$2.729.a_acec
2.9.a_j$3$2.729.a_acec
2.9.j_bk$3$2.729.a_acec
2.9.ad_s$4$(not in LMFDB)
2.9.d_s$4$(not in LMFDB)
2.9.am_cc$6$(not in LMFDB)
2.9.ag_bb$6$(not in LMFDB)
2.9.g_bb$6$(not in LMFDB)
2.9.j_bk$6$(not in LMFDB)
2.9.m_cc$6$(not in LMFDB)
2.9.ag_s$12$(not in LMFDB)
2.9.a_aj$12$(not in LMFDB)
2.9.a_s$12$(not in LMFDB)
2.9.g_s$12$(not in LMFDB)
2.9.a_a$24$(not in LMFDB)
2.9.ad_j$30$(not in LMFDB)
2.9.d_j$30$(not in LMFDB)