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

• Label: 2.73.bg_pm
• 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:	$( 1 + 16 x + 73 x^{2} )^{2}$
	$1 + 32 x + 402 x^{2} + 2336 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.885799748780$, $\pm0.885799748780$
Angle rank:	$1$ (numerical)
Jacobians:	$5$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8100$	$27248400$	$151795952100$	$806378250240000$	$4297542314347102500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$106$	$5110$	$390202$	$28395358$	$2073031306$	$151335081430$	$11047387777882$	$806460201328318$	$58871585741428906$	$4297625837184282550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

• $y^2=x^6+x^3+24$
• $y^2=18 x^6+23 x^4+23 x^2+18$
• $y^2=8 x^6+17 x^5+37 x^4+14 x^3+48 x^2+14 x+27$
• $y^2=x^6+x^3+65$
• $y^2=29 x^6+21 x^5+50 x^4+32 x^3+9 x^2+58 x+14$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.q^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.abg_pm	$2$	(not in LMFDB)
2.73.a_aeg	$2$	(not in LMFDB)
2.73.aq_hb	$3$	(not in LMFDB)