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

• Label: 2.73.abb_mk
• 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:	$( 1 - 16 x + 73 x^{2} )( 1 - 11 x + 73 x^{2} )$
	$1 - 27 x + 322 x^{2} - 1971 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.114200251220$, $\pm0.277387524567$
Angle rank:	$2$ (numerical)
Jacobians:	$25$
Isomorphism classes:	165

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3654$	$27953100$	$151523428896$	$806704103520000$	$4297733246546141094$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$47$	$5245$	$389504$	$28406833$	$2073123407$	$151334269810$	$11047397353319$	$806460102746593$	$58871587186332992$	$4297625836587137725$

Jacobians and polarizations

This isogeny class contains the Jacobians of 25 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=70x^6+56x^5+68x^4+23x^3+66x^2+62x+5$
• $y^2=36x^6+32x^5+10x^4+52x^3+20x^2+49x+59$
• $y^2=39x^6+20x^5+57x^4+64x^3+46x^2+43x+34$
• $y^2=10x^6+68x^5+32x^4+68x^3+24x^2+69x+51$
• $y^2=57x^6+39x^5+70x^4+x^3+19x^2+17x+37$
• $y^2=39x^6+34x^5+72x^4+62x^3+14x^2+6x+19$
• $y^2=41x^6+23x^5+64x^4+24x^3+66x^2+18x+9$
• $y^2=17x^6+25x^5+2x^4+33x^3+9x^2+41x+34$
• $y^2=47x^6+6x^5+35x^4+49x^3+40x^2+59x+51$
• $y^2=21x^6+17x^5+67x^4+27x^3+29x^2+69x+59$
• $y^2=39x^6+6x^5+46x^4+52x^3+64x^2+22x+42$
• $y^2=3x^6+42x^5+43x^4+6x^3+22x^2+2x+11$
• $y^2=36x^6+16x^5+36x^4+3x^3+45x^2+15x+29$
• $y^2=55x^6+55x^5+45x^4+64x^3+5x^2+56x+7$
• $y^2=64x^6+7x^5+x^4+x^3+33x^2+29x+71$
• $y^2=10x^6+13x^5+15x^4+61x^3+17x^2+6x+9$
• $y^2=28x^6+29x^5+47x^4+13x^3+38x^2+12x+61$
• $y^2=52x^6+19x^5+x^3+61x^2+8x+22$
• $y^2=37x^6+49x^5+2x^4+30x^3+58x^2+39x+52$
• $y^2=51x^6+53x^5+58x^4+6x^3+13x^2+4x+10$
• $y^2=56x^6+51x^5+5x^4+5x^3+51x^2+7x+33$
• $y^2=57x^6+20x^5+58x^4+65x^3+69x^2+39x+56$
• $y^2=43x^6+57x^5+17x^4+42x^3+64x^2+66x+34$
• $y^2=34x^6+67x^5+56x^4+27x^3+19x^2+61x+13$
• $y^2=68x^6+59x^5+51x^4+21x^3+62x^2+25x+57$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.aq $\times$ 1.73.al 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.aq : \(\Q(\sqrt{-1}) \).
• 1.73.al : \(\Q(\sqrt{-19}) \).

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.af_abe	$2$	(not in LMFDB)
2.73.f_abe	$2$	(not in LMFDB)
2.73.bb_mk	$2$	(not in LMFDB)