Abelian variety isogeny class 2.61.abe_nj over $\F_{61}$

• Label: 2.61.abe_nj
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 - 15 x + 61 x^{2} )^{2}$
	$1 - 30 x + 347 x^{2} - 1830 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0900194921159$, $\pm0.0900194921159$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2209$	$13097161$	$51235227904$	$191619651155625$	$713327584011549529$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$3516$	$225722$	$13839508$	$844578152$	$51520488486$	$3142745655032$	$191707348320868$	$11694146450728322$	$713342914876556556$

Jacobians and polarizations

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

• $y^2=29x^6+26x^5+50x^4+46x^3+6x^2+40x+26$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ap^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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
2.61.a_adz	$2$	(not in LMFDB)
2.61.be_nj	$2$	(not in LMFDB)
2.61.p_gi	$3$	(not in LMFDB)