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

• Label: 2.71.aba_ly
• Base field: $\F_{71}$
• 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:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$( 1 - 14 x + 71 x^{2} )( 1 - 12 x + 71 x^{2} )$
	$1 - 26 x + 310 x^{2} - 1846 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.187913521440$, $\pm0.247758306964$
Angle rank:	$2$ (numerical)
Jacobians:	$24$
Isomorphism classes:	64

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3480$	$25139520$	$128482731000$	$646191853332480$	$3255506208686787000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$4986$	$358978$	$25428926$	$1804374926$	$128100973338$	$9095119413986$	$645753479974846$	$45848500154994478$	$3255243547404527226$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=9x^6+57x^5+50x^4+45x^3+6x^2+9x+6$
• $y^2=14x^6+34x^5+7x^4+2x^3+7x^2+34x+14$
• $y^2=34x^6+3x^5+x^4+22x^3+57x^2+20x+69$
• $y^2=12x^6+68x^5+31x^4+47x^3+31x^2+68x+12$
• $y^2=67x^6+5x^5+53x^4+23x^3+62x^2+19x+35$
• $y^2=55x^6+70x^5+24x^4+52x^3+24x^2+70x+55$
• $y^2=56x^6+56x^4+15x^3+17x^2+13$
• $y^2=32x^6+4x^5+x^4+39x^3+60x^2+58x+8$
• $y^2=12x^6+47x^5+48x^4+19x^3+48x^2+47x+12$
• $y^2=5x^6+25x^5+56x^4+65x^3+56x^2+25x+5$
• $y^2=10x^6+21x^5+41x^4+53x^3+42x^2+44x+18$
• $y^2=37x^6+53x^5+41x^4+18x^3+52x^2+51x+5$
• $y^2=67x^6+45x^5+20x^4+63x^3+32x^2+30x+62$
• $y^2=58x^6+45x^5+13x^4+11x^3+13x^2+45x+58$
• $y^2=56x^6+33x^5+12x^4+47x^3+12x^2+33x+56$
• $y^2=51x^6+55x^5+2x^4+2x^2+55x+51$
• $y^2=27x^6+15x^5+6x^4+23x^3+6x^2+15x+27$
• $y^2=53x^6+34x^5+16x^4+33x^3+16x^2+34x+53$
• $y^2=53x^6+39x^5+44x^4+70x^3+13x^2+33x+31$
• $y^2=53x^6+41x^4+17x^3+41x^2+53$
• $y^2=23x^6+12x^5+68x^4+44x^3+52x^2+8x+34$
• $y^2=64x^6+31x^5+26x^4+67x^3+11x^2+63x+36$
• $y^2=68x^6+51x^5+60x^4+8x^3+60x^2+51x+68$
• $y^2=51x^6+69x^5+40x^4+19x^3+60x^2+31x+39$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ao $\times$ 1.71.am and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.71.ao : \(\Q(\sqrt{-22}) \).
• 1.71.am : \(\Q(\sqrt{-35}) \).

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.ac_aba	$2$	(not in LMFDB)
2.71.c_aba	$2$	(not in LMFDB)
2.71.ba_ly	$2$	(not in LMFDB)