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

• Label: 2.97.abj_te
• Base field: $\F_{97}$
• 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:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$( 1 - 19 x + 97 x^{2} )( 1 - 16 x + 97 x^{2} )$
	$1 - 35 x + 498 x^{2} - 3395 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0849741350078$, $\pm0.198227810371$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	32

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6478$	$86403564$	$832270326496$	$7837956223958016$	$73743631456904466718$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$9181$	$911904$	$88535185$	$8587482183$	$832973573122$	$80798295971799$	$7837433641990369$	$760231058585818848$	$73742412687641168461$

Jacobians and polarizations

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

• $y^2=35x^6+75x^5+72x^4+21x^3+47x^2+74x+23$
• $y^2=15x^6+38x^5+56x^4+51x^3+x^2+21x+29$
• $y^2=68x^6+63x^5+81x^4+8x^3+x^2+61x+72$
• $y^2=53x^6+15x^5+20x^4+43x^3+65x^2+18x+72$
• $y^2=85x^6+15x^5+62x^4+61x^3+90x^2+62x+80$
• $y^2=68x^6+91x^5+80x^4+33x^3+54x^2+76x+39$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.at $\times$ 1.97.aq and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.97.at : \(\Q(\sqrt{-3}) \).
• 1.97.aq : \(\Q(\sqrt{-33}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.97.ad_aeg	$2$	(not in LMFDB)
2.97.d_aeg	$2$	(not in LMFDB)
2.97.bj_te	$2$	(not in LMFDB)
2.97.al_ek	$3$	(not in LMFDB)
2.97.ac_abe	$3$	(not in LMFDB)