Abelian variety isogeny class 2.53.ba_kp over $\F_{53}$

• Label: 2.53.ba_kp
• Base field: $\F_{53}$
• 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:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$( 1 + 13 x + 53 x^{2} )^{2}$
	$1 + 26 x + 275 x^{2} + 1378 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.851293248891$, $\pm0.851293248891$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4489$	$7546009$	$22203384064$	$62285731721161$	$174863779556805889$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$2684$	$149138$	$7893780$	$418138840$	$22164922838$	$1174706840296$	$62259716534884$	$3299763480072074$	$174887470433457164$

Jacobians and polarizations

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

• $y^2=42 x^6+5 x^5+2 x^4+49 x^3+41 x^2+21 x+44$
• $y^2=31 x^6+43 x^5+8 x^4+47 x^3+38 x^2+34 x+37$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53}$.

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.n^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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
2.53.aba_kp	$2$	(not in LMFDB)
2.53.a_acl	$2$	(not in LMFDB)
2.53.an_em	$3$	(not in LMFDB)