Abelian variety isogeny class 2.169.abt_bgm over $\F_{13^{2}}$

• Label: 2.169.abt_bgm
• Base field: $\F_{13^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{13^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 23 x + 169 x^{2} )( 1 - 22 x + 169 x^{2} )$
	$1 - 45 x + 844 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.154420958311$, $\pm0.178912375022$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21756$	$806190336$	$23298094520064$	$665462657105777664$	$19005137818449437301276$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$125$	$28225$	$4826810$	$815787169$	$137859754325$	$23298103917646$	$3937376595232205$	$665416610737982209$	$112455406951957393130$	$19004963774625235380625$

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 $\F_{13^{12}}$.

Endomorphism algebra over $\F_{13^{2}}$

The isogeny class factors as 1.169.ax $\times$ 1.169.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.169.ax : \(\Q(\sqrt{-3}) \).
• 1.169.aw : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{13^{2}}$

The base change of $A$ to $\F_{13^{12}}$ is 1.23298085122481.uortm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{13^{4}}$
  The base change of $A$ to $\F_{13^{4}}$ is 1.28561.ahj $\times$ 1.28561.afq. The endomorphism algebra for each factor is:

  • 1.28561.ahj : \(\Q(\sqrt{-3}) \).
  • 1.28561.afq : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{13^{6}}$
  The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.atm $\times$ 1.4826809.tm. The endomorphism algebra for each factor is:

  • 1.4826809.atm : \(\Q(\sqrt{-3}) \).
  • 1.4826809.tm : \(\Q(\sqrt{-3}) \).

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
2.169.ab_agm	$2$	(not in LMFDB)
2.169.b_agm	$2$	(not in LMFDB)
2.169.bt_bgm	$2$	(not in LMFDB)
2.169.ay_nx	$3$	(not in LMFDB)
2.169.av_me	$3$	(not in LMFDB)
2.169.a_ahj	$3$	(not in LMFDB)
2.169.a_afq	$3$	(not in LMFDB)
2.169.a_mz	$3$	(not in LMFDB)
2.169.v_me	$3$	(not in LMFDB)
2.169.y_nx	$3$	(not in LMFDB)