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

• Label: 2.53.abc_lq
• 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:	$( 1 - 14 x + 53 x^{2} )^{2}$
	$1 - 28 x + 302 x^{2} - 1484 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0885855327829$, $\pm0.0885855327829$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1600$	$7398400$	$22010689600$	$62220544000000$	$174881370619240000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$2630$	$147842$	$7885518$	$418180906$	$22164419990$	$1174712737042$	$62259709652638$	$3299763776528186$	$174887471931890150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=25 x^6+42 x^4+42 x^2+25$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ao^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_adm	$2$	(not in LMFDB)
2.53.bc_lq	$2$	(not in LMFDB)
2.53.o_fn	$3$	(not in LMFDB)