Abelian variety isogeny class 2.243.acd_bvu over $\F_{3^{5}}$

• Label: 2.243.acd_bvu
• Base field: $\F_{3^{5}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{5}}$
Dimension:	$2$
L-polynomial:	$( 1 - 28 x + 243 x^{2} )( 1 - 27 x + 243 x^{2} )$
	$1 - 55 x + 1242 x^{2} - 13365 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.144947286894$, $\pm0.166666666667$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$46872$	$3455028864$	$205869063474144$	$12157973500754704896$	$717900350592743310476232$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$189$	$58509$	$14347368$	$3486872745$	$847291398219$	$205891187118678$	$50031545933302353$	$12157665468659143569$	$2954312706613473384984$	$717897987691273161799389$

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 $\F_{3^{30}}$.

Endomorphism algebra over $\F_{3^{5}}$

The isogeny class factors as 1.243.abc $\times$ 1.243.abb and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.243.abc : \(\Q(\sqrt{-47}) \).
• 1.243.abb : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{3^{5}}$

The base change of $A$ to $\F_{3^{30}}$ is 1.205891132094649.cfpwcs $\times$ 1.205891132094649.ckuukc. The endomorphism algebra for each factor is:

• 1.205891132094649.cfpwcs : \(\Q(\sqrt{-47}) \).
• 1.205891132094649.ckuukc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{10}}$
  The base change of $A$ to $\F_{3^{10}}$ is 1.59049.alm $\times$ 1.59049.ajj. The endomorphism algebra for each factor is:

  • 1.59049.alm : \(\Q(\sqrt{-47}) \).
  • 1.59049.ajj : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{3^{15}}$
  The base change of $A$ to $\F_{3^{15}}$ is 1.14348907.achg $\times$ 1.14348907.a. The endomorphism algebra for each factor is:

  • 1.14348907.achg : \(\Q(\sqrt{-47}) \).
  • 1.14348907.a : \(\Q(\sqrt{-3}) \).

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
2.243.ab_akk	$2$	(not in LMFDB)
2.243.b_akk	$2$	(not in LMFDB)
2.243.cd_bvu	$2$	(not in LMFDB)
2.243.abc_ss	$3$	(not in LMFDB)