Abelian variety isogeny class 2.71.aba_lp over $\F_{71}$

• Label: 2.71.aba_lp
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 - 26 x + 301 x^{2} - 1846 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0914029226447$, $\pm0.301580343135$
Angle rank:	$2$ (numerical)
Number field:	4.0.6824000.2
Galois group:	$D_{4}$
Jacobians:	$18$
Isomorphism classes:	36

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3471$	$25043265$	$128230388676$	$645852806895465$	$3255219114551654991$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$4968$	$358276$	$25415588$	$1804215806$	$128099800998$	$9095117055266$	$645753551228548$	$45848501348850076$	$3255243558091539048$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=47x^6+43x^5+66x^4+66x^3+39x^2+35x+48$
• $y^2=61x^6+68x^5+2x^4+46x^3+4x^2+46x+39$
• $y^2=56x^6+24x^5+22x^4+25x^3+23x^2+52x+61$
• $y^2=43x^6+69x^5+56x^4+21x^3+70x^2+43x+66$
• $y^2=34x^6+4x^5+50x^4+x^3+22x^2+31x+62$
• $y^2=17x^6+60x^5+46x^4+49x^3+49x^2+7x+51$
• $y^2=49x^6+11x^5+17x^4+17x^3+10x^2+41x+46$
• $y^2=61x^6+20x^5+41x^4+13x^3+64x^2+70x+14$
• $y^2=60x^6+28x^5+24x^4+54x^3+19x^2+27x+60$
• $y^2=54x^6+64x^5+34x^4+26x^3+45x^2+61x+67$
• $y^2=53x^6+27x^4+33x^3+20x^2+21x+56$
• $y^2=38x^6+34x^5+29x^4+64x^3+43x^2+42x+18$
• $y^2=19x^6+21x^5+25x^4+38x^3+47x^2+13x+23$
• $y^2=21x^6+57x^5+66x^4+33x^3+5x^2+11x+62$
• $y^2=17x^6+39x^5+64x^4+14x^3+21x^2+69x+67$
• $y^2=59x^6+69x^5+70x^4+3x^3+60x^2+37x+25$
• $y^2=35x^6+15x^5+61x^4+23x^3+60x^2+59$
• $y^2=50x^6+56x^5+56x^4+45x^3+25x^2+20x+65$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is 4.0.6824000.2.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.71.ba_lp	$2$	(not in LMFDB)