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

• Label: 2.97.abm_vj
• 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:	$( 1 - 19 x + 97 x^{2} )^{2}$
	$1 - 38 x + 555 x^{2} - 3686 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0849741350078$, $\pm0.0849741350078$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6241$	$85433049$	$830547886336$	$7835827755484521$	$73741668375066892561$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$9076$	$910014$	$88511140$	$8587253580$	$832972117822$	$80798295030828$	$7837433783928004$	$760231061232421758$	$73742412720085649236$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=5 x^6+5 x^3+59$
• $y^2=15 x^6+94 x^5+20 x^4+69 x^3+90 x^2+12 x+21$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.a_agl	$2$	(not in LMFDB)
2.97.bm_vj	$2$	(not in LMFDB)
2.97.ao_dv	$3$	(not in LMFDB)
2.97.af_acu	$3$	(not in LMFDB)
2.97.k_il	$3$	(not in LMFDB)
2.97.t_ke	$3$	(not in LMFDB)
2.97.bc_pa	$3$	(not in LMFDB)