Abelian variety isogeny class 2.81.abb_mm over $\F_{3^{4}}$

• Label: 2.81.abb_mm
• Base field: $\F_{3^{4}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x )^{2}( 1 - 9 x + 81 x^{2} )$
	$1 - 27 x + 324 x^{2} - 2187 x^{3} + 6561 x^{4}$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4672$	$42515200$	$282428473600$	$1852737759308800$	$12157047799607605312$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$6481$	$531442$	$43040161$	$3486607255$	$282427410718$	$22876778106055$	$1853020145805121$	$150094635296999122$	$12157665455570144401$

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 $\F_{3^{24}}$.

Endomorphism algebra over $\F_{3^{4}}$

The isogeny class factors as 1.81.as $\times$ 1.81.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.81.as : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.81.aj : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{3^{4}}$

The base change of $A$ to $\F_{3^{24}}$ is 1.282429536481.acimic^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

• Endomorphism algebra over $\F_{3^{8}}$
  The base change of $A$ to $\F_{3^{8}}$ is 1.6561.agg $\times$ 1.6561.dd. The endomorphism algebra for each factor is:

  • 1.6561.agg : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 1.6561.dd : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{3^{12}}$
  The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.cec. The endomorphism algebra for each factor is:

  • 1.531441.acec : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 1.531441.cec : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.81.aj_a	$2$	(not in LMFDB)
2.81.j_a	$2$	(not in LMFDB)
2.81.bb_mm	$2$	(not in LMFDB)
2.81.a_agg	$3$	(not in LMFDB)
2.81.a_dd	$3$	(not in LMFDB)