Abelian variety isogeny class 2.13.ai_bm over $\F_{13}$

• Label: 2.13.ai_bm
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$( 1 - 6 x + 13 x^{2} )( 1 - 2 x + 13 x^{2} )$
	$1 - 8 x + 38 x^{2} - 104 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.187167041811$, $\pm0.410543812489$
Angle rank:	$2$ (numerical)
Jacobians:	$12$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$96$	$30720$	$5025888$	$818380800$	$137854828896$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$182$	$2286$	$28654$	$371286$	$4830374$	$62772030$	$815767006$	$10604268198$	$137857106582$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=5 x^6+4 x^5+6 x^4+7 x^3+2 x^2+12 x+5$
• $y^2=4 x^6+10 x^5+3 x^4+3 x^3+4 x^2+12 x+9$
• $y^2=5 x^6+6 x^5+11 x^4+3 x^3+11 x^2+6 x+5$
• $y^2=2 x^6+9 x^5+x^4+10 x^3+11 x^2+6 x+2$
• $y^2=7 x^5+3 x^4+4 x^3+3 x^2+7 x$
• $y^2=2 x^6+9 x^5+3 x^4+4 x^3+3 x^2+9 x+2$
• $y^2=11 x^6+4 x^5+7 x^4+5 x^3+11 x^2+x+2$
• $y^2=6 x^6+10 x^5+2 x^4+7 x^3+2 x^2+10 x+6$
• $y^2=8 x^6+8 x^5+9 x^4+9 x^3+3 x^2+2 x+5$
• $y^2=4 x^6+10 x^5+11 x^4+11 x^3+2 x^2+10 x+9$
• $y^2=2 x^6+3 x^5+12 x^4+2 x^3+12 x^2+3 x+2$
• $y^2=8 x^6+5 x^5+11 x^4+8 x^3+11 x^2+11 x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ag $\times$ 1.13.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.13.ag : \(\Q(\sqrt{-1}) \).
• 1.13.ac : \(\Q(\sqrt{-3}) \).

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
2.13.ae_o	$2$	2.169.m_eo
2.13.e_o	$2$	2.169.m_eo
2.13.i_bm	$2$	2.169.m_eo
2.13.al_ce	$3$	(not in LMFDB)
2.13.b_aq	$3$	(not in LMFDB)