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

• Label: 2.73.a_afn
• 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:	$( 1 - 17 x + 73 x^{2} )( 1 + 17 x + 73 x^{2} )$
	$1 - 143 x^{2} + 5329 x^{4}$
Frobenius angles:	$\pm0.0323195869136$, $\pm0.967680413086$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
Isomorphism classes:	98
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5187$	$26904969$	$151333588224$	$805904150179401$	$4297625827517201907$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5044$	$389018$	$28378660$	$2073071594$	$151332950158$	$11047398519098$	$806460013759684$	$58871586708267914$	$4297625825330846164$

Jacobians and polarizations

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

• $y^2=x^6+59 x^3+30$
• $y^2=x^6+47 x^3+17$
• $y^2=x^6+x^3+7$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{2}}$.

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ar $\times$ 1.73.r 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.r : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{73}$

The base change of $A$ to $\F_{73^{2}}$ is 1.5329.afn^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
2.73.abi_qt	$2$	(not in LMFDB)
2.73.bi_qt	$2$	(not in LMFDB)
2.73.abb_me	$3$	(not in LMFDB)
2.73.ay_kf	$3$	(not in LMFDB)
2.73.ad_cy	$3$	(not in LMFDB)
2.73.a_bu	$3$	(not in LMFDB)
2.73.a_dt	$3$	(not in LMFDB)
2.73.d_cy	$3$	(not in LMFDB)
2.73.y_kf	$3$	(not in LMFDB)
2.73.bb_me	$3$	(not in LMFDB)