Abelian variety isogeny class 2.73.abf_ou over $\F_{73}$

• Label: 2.73.abf_ou
• Base field: $\F_{73}$
• 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:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$( 1 - 17 x + 73 x^{2} )( 1 - 14 x + 73 x^{2} )$
	$1 - 31 x + 384 x^{2} - 2263 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.0323195869136$, $\pm0.194368965322$
Angle rank:	$2$ (numerical)
Jacobians:	$3$
Isomorphism classes:	18

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3420$	$27387360$	$150996953520$	$806413694486400$	$4297648897845359100$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$5137$	$388150$	$28396609$	$2073082723$	$151334262574$	$11047396300459$	$806460043070401$	$58871586069495430$	$4297625823435766657$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=63x^6+62x^5+56x^4+68x^3+53x^2+21x+71$
• $y^2=34x^6+53x^5+24x^4+37x^3+56x^2+6x+62$
• $y^2=42x^6+25x^5+30x^4+14x^3+44x^2+15x+11$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73}$.

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ar $\times$ 1.73.ao and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.73.ar : \(\Q(\sqrt{-3}) \).
• 1.73.ao : \(\Q(\sqrt{-6}) \).

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
2.73.ad_ado	$2$	(not in LMFDB)
2.73.d_ado	$2$	(not in LMFDB)
2.73.bf_ou	$2$	(not in LMFDB)
2.73.ah_bw	$3$	(not in LMFDB)
2.73.ae_g	$3$	(not in LMFDB)