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

• Label: 2.71.abc_mv
• 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:	$1 - 28 x + 333 x^{2} - 1988 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0863455308586$, $\pm0.254046449771$
Angle rank:	$2$ (numerical)
Number field:	4.0.74225.1
Galois group:	$D_{4}$
Jacobians:	$14$
Isomorphism classes:	14

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3359$	$24826369$	$128120321600$	$645889792113689$	$3255312208138660759$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$4924$	$357968$	$25417044$	$1804267404$	$128100269998$	$9095117474564$	$645753509352804$	$45848500779659888$	$3255243554748881004$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=62x^6+2x^5+69x^4+49x^3+47x^2+64x+61$
• $y^2=63x^6+22x^5+52x^4+40x^3+23x^2+8x+31$
• $y^2=47x^6+21x^5+66x^4+16x^3+53x^2+27x+20$
• $y^2=13x^6+40x^5+50x^4+57x^3+51x^2+38x+70$
• $y^2=55x^6+43x^5+53x^4+32x^3+40x^2+56x+38$
• $y^2=35x^6+16x^5+11x^4+46x^3+54x^2+28x+42$
• $y^2=55x^6+45x^5+22x^4+19x^3+37x^2+22x+7$
• $y^2=70x^6+35x^5+23x^4+68x^3+2x^2+35x+42$
• $y^2=55x^6+15x^5+42x^4+53x^3+21x^2+54x+31$
• $y^2=14x^6+25x^5+28x^4+43x^3+23x^2+34x+44$
• $y^2=38x^6+2x^5+46x^3+44x^2+25x+37$
• $y^2=37x^6+61x^5+57x^3+17x^2+8x+43$
• $y^2=44x^6+17x^5+26x^4+35x^3+62x^2+65x+7$
• $y^2=41x^6+57x^5+2x^4+14x^3+63x^2+46x+2$

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 4.0.74225.1.

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.bc_mv	$2$	(not in LMFDB)