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

• Label: 2.97.abj_tg
• 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: no

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$( 1 - 18 x + 97 x^{2} )( 1 - 17 x + 97 x^{2} )$
	$1 - 35 x + 500 x^{2} - 3395 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.133124938748$, $\pm0.168554845458$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	12

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6480$	$86443200$	$832462712640$	$7838470556640000$	$73744605284354888400$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$9185$	$912114$	$88540993$	$8587595583$	$832975301630$	$80798317227999$	$7837433848912513$	$760231059978310578$	$73742412688723347425$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.as $\times$ 1.97.ar 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.as : \(\Q(\sqrt{-1}) \).
• 1.97.ar : \(\Q(\sqrt{-11}) \).

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.ab_aei	$2$	(not in LMFDB)
2.97.b_aei	$2$	(not in LMFDB)
2.97.bj_tg	$2$	(not in LMFDB)