Abelian variety isogeny class 2.97.abf_qe over $\F_{97}$

• Label: 2.97.abf_qe
• Base field: $\F_{97}$
• 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:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 31 x + 420 x^{2} - 3007 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0660699152030$, $\pm0.297052668424$
Angle rank:	$2$ (numerical)
Number field:	4.0.3352968.4
Galois group:	$D_{4}$
Jacobians:	$24$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6792$	$87399456$	$833197379616$	$7837692698288256$	$73741677706819769352$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$67$	$9289$	$912922$	$88532209$	$8587254667$	$832970006494$	$80798264942875$	$7837433543215393$	$760231060069385626$	$73742412714900938329$

Jacobians and polarizations

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

• $y^2=57x^6+87x^5+96x^4+85x^3+2x^2+17x+36$
• $y^2=59x^6+12x^5+80x^4+11x^3+53x^2+80x+19$
• $y^2=82x^6+4x^5+58x^4+60x^3+8x^2+64x+24$
• $y^2=28x^6+77x^5+11x^4+64x^3+46x^2+65x+37$
• $y^2=75x^6+17x^5+46x^4+87x^3+70x^2+17x+46$
• $y^2=84x^6+25x^5+69x^4+12x^3+96x^2+48x+15$
• $y^2=42x^6+37x^5+95x^4+42x^3+18x^2+24x+74$
• $y^2=39x^6+50x^5+70x^4+47x^3+61x^2+22x+23$
• $y^2=8x^5+83x^4+37x^3+41x^2+21x+82$
• $y^2=66x^6+61x^5+20x^4+40x^3+39x^2+17x+31$
• $y^2=59x^6+47x^5+85x^4+10x^3+51x^2+90x$
• $y^2=70x^6+54x^5+80x^4+20x^3+60x^2+83x+38$
• $y^2=10x^6+6x^5+40x^4+68x^3+30x^2+12x+20$
• $y^2=50x^6+96x^5+64x^4+27x^3+23x^2+39x+20$
• $y^2=5x^6+32x^5+95x^4+80x^3+31x^2+22x+42$
• $y^2=57x^6+38x^5+78x^4+30x^3+23x^2+37x+92$
• $y^2=80x^6+93x^5+76x^4+82x^3+40x^2+7x+62$
• $y^2=96x^6+70x^5+94x^4+2x^3+81x^2+69x+34$
• $y^2=83x^6+37x^5+91x^4+76x^3+11x^2+76x+24$
• $y^2=23x^6+13x^5+56x^4+8x^3+20x^2+22x+75$
• $y^2=63x^6+49x^5+58x^4+33x^3+87x^2+32x+61$
• $y^2=37x^6+45x^5+73x^4+27x^3+24x^2+38x+61$
• $y^2=46x^6+93x^5+46x^4+71x^3+73x^2+48x+25$
• $y^2=31x^6+71x^5+4x^4+23x^3+43x^2+46x+83$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97}$.

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is 4.0.3352968.4.

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
2.97.bf_qe	$2$	(not in LMFDB)