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

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

Invariants

Base field:	$\F_{3^{5}}$
Dimension:	$2$
L-polynomial:	$( 1 - 31 x + 243 x^{2} )( 1 - 27 x + 243 x^{2} )$
	$1 - 58 x + 1323 x^{2} - 14094 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.0339262533067$, $\pm0.166666666667$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$46221$	$3444620025$	$205787963446128$	$12157496427346895625$	$717897995013287241115221$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$186$	$58332$	$14341716$	$3486735924$	$847288618086$	$205891137765414$	$50031545157902130$	$12157665457955226276$	$2954312706488420494188$	$717897987690185493145932$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{30}}$.

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

The isogeny class factors as 1.243.abf $\times$ 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.abf : \(\Q(\sqrt{-11}) \).
• 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.abykdru $\times$ 1.205891132094649.ckuukc. The endomorphism algebra for each factor is:

• 1.205891132094649.abykdru : \(\Q(\sqrt{-11}) \).
• 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.ash $\times$ 1.59049.ajj. The endomorphism algebra for each factor is:

  • 1.59049.ash : \(\Q(\sqrt{-11}) \).
  • 1.59049.ajj : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{3^{15}}$
  The base change of $A$ to $\F_{3^{15}}$ is 1.14348907.akqq $\times$ 1.14348907.a. The endomorphism algebra for each factor is:

  • 1.14348907.akqq : \(\Q(\sqrt{-11}) \).
  • 1.14348907.a : \(\Q(\sqrt{-3}) \).

Base change

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

Subfield	Primitive Model
$\F_{3}$	2.3.c_d

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.243.ae_ann	$2$	(not in LMFDB)
2.243.e_ann	$2$	(not in LMFDB)
2.243.cg_byx	$2$	(not in LMFDB)
2.243.abf_ss	$3$	(not in LMFDB)
2.243.ae_ann	$3$	(not in LMFDB)