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

• Label: 2.97.abk_tx
• 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:	$( 1 - 19 x + 97 x^{2} )( 1 - 17 x + 97 x^{2} )$
	$1 - 36 x + 517 x^{2} - 3492 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0849741350078$, $\pm0.168554845458$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	18

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6399$	$86098545$	$831790959552$	$7837497600744825$	$73743441840801198279$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$9148$	$911378$	$88530004$	$8587460102$	$832973885566$	$80798304936950$	$7837433770308196$	$760231059850440146$	$73742412695335226668$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+87x^5+29x^4+24x^3+29x^2+87x+2$
• $y^2=37x^6+63x^5+52x^4+55x^3+52x^2+63x+37$
• $y^2=67x^6+65x^5+38x^4+39x^3+38x^2+65x+67$
• $y^2=67x^6+62x^5+45x^4+47x^3+45x^2+62x+67$
• $y^2=13x^6+51x^5+58x^4+63x^3+58x^2+51x+13$
• $y^2=40x^6+61x^5+60x^4+51x^3+60x^2+61x+40$
• $y^2=30x^6+17x^5+11x^4+53x^3+11x^2+17x+30$
• $y^2=41x^6+26x^5+80x^4+14x^3+80x^2+26x+41$
• $y^2=21x^6+36x^5+81x^4+65x^3+81x^2+36x+21$
• $y^2=92x^6+62x^5+76x^4+80x^3+76x^2+62x+92$
• $y^2=28x^6+18x^5+24x^3+18x+28$
• $y^2=13x^6+67x^5+12x^4+94x^3+12x^2+67x+13$

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.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.at : \(\Q(\sqrt{-3}) \).
• 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.ac_aez	$2$	(not in LMFDB)
2.97.c_aez	$2$	(not in LMFDB)
2.97.bk_tx	$2$	(not in LMFDB)
2.97.am_ef	$3$	(not in LMFDB)
2.97.ad_abs	$3$	(not in LMFDB)