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

• Label: 2.71.aba_lv
• 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:	$( 1 - 15 x + 71 x^{2} )( 1 - 11 x + 71 x^{2} )$
	$1 - 26 x + 307 x^{2} - 1846 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.150643965450$, $\pm0.273623649113$
Angle rank:	$2$ (numerical)
Jacobians:	$18$
Isomorphism classes:	30

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3477$	$25107417$	$128398600368$	$646079746379625$	$3255414731148905877$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$4980$	$358744$	$25424516$	$1804324226$	$128100659022$	$9095120282126$	$645753529748356$	$45848500855704904$	$3255243553375771380$

Jacobians and polarizations

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

• $y^2=35x^6+16x^5+4x^4+60x^3+4x^2+16x+35$
• $y^2=56x^6+58x^5+3x^4+63x^3+3x^2+58x+56$
• $y^2=62x^6+34x^5+41x^4+36x^3+49x^2+53x+7$
• $y^2=50x^6+47x^5+32x^4+67x^3+31x^2+6x+35$
• $y^2=52x^6+21x^5+63x^4+35x^2+68x+5$
• $y^2=13x^6+9x^5+65x^4+30x^3+65x^2+9x+13$
• $y^2=63x^6+20x^5+58x^4+44x^3+58x^2+20x+63$
• $y^2=67x^6+41x^5+51x^4+19x^3+51x^2+41x+67$
• $y^2=11x^6+9x^5+16x^4+16x^3+16x^2+9x+11$
• $y^2=14x^6+2x^5+16x^4+38x^3+70x^2+70x+69$
• $y^2=25x^6+68x^4+33x^3+43x^2+23x+53$
• $y^2=39x^6+41x^5+67x^4+43x^3+50x^2+60x+61$
• $y^2=3x^6+4x^5+37x^4+63x^3+4x^2+51x+45$
• $y^2=28x^6+31x^5+23x^4+47x^3+23x^2+31x+28$
• $y^2=43x^6+35x^5+30x^4+57x^3+11x^2+22x+42$
• $y^2=7x^6+26x^5+12x^4+12x^3+12x^2+26x+7$
• $y^2=67x^6+37x^5+60x^4+62x^3+52x^2+6x+65$
• $y^2=67x^6+22x^5+47x^4+10x^3+47x^2+22x+67$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

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

• 1.71.ap : \(\Q(\sqrt{-59}) \).
• 1.71.al : \(\Q(\sqrt{-163}) \).

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.ae_ax	$2$	(not in LMFDB)
2.71.e_ax	$2$	(not in LMFDB)
2.71.ba_lv	$2$	(not in LMFDB)