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

• Label: 2.97.abk_tw
• 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:	$1 - 36 x + 516 x^{2} - 3492 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0540701476596$, $\pm0.181368932438$
Angle rank:	$2$ (numerical)
Number field:	4.0.80128.1
Galois group:	$D_{4}$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6398$	$86078692$	$831691995806$	$7837221476059024$	$73742885366502071678$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$9146$	$911270$	$88526886$	$8587395302$	$832972804058$	$80798289687422$	$7837433584319614$	$760231057877143742$	$73742412677367152666$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=94x^6+30x^5+56x^4+9x^3+89x^2+49x+89$
• $y^2=7x^6+2x^5+45x^4+67x^3+37x^2+34x+60$
• $y^2=2x^6+74x^5+71x^4+61x^3+62x^2+22x+9$
• $y^2=23x^6+14x^5+41x^4+6x^3+17x^2+72x+81$
• $y^2=15x^6+30x^5+87x^4+68x^3+83x^2+41x+55$
• $y^2=55x^6+82x^5+96x^4+27x^3+68x^2+32x+11$

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 4.0.80128.1.

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
2.97.bk_tw	$2$	(not in LMFDB)