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

• Label: 2.61.aba_lf
• 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:	$( 1 - 13 x + 61 x^{2} )^{2}$
	$1 - 26 x + 291 x^{2} - 1586 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.187058313935$, $\pm0.187058313935$
Angle rank:	$1$ (numerical)
Jacobians:	$8$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2401$	$13505625$	$51603482896$	$191852278655625$	$713439077260126441$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3628$	$227346$	$13856308$	$844710156$	$51521216038$	$3142746832716$	$191707313612068$	$11694145857028026$	$713342908559901148$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=2 x^6+2 x^3+2$
• $y^2=29 x^6+53 x^5+28 x^4+29 x^3+7 x^2+30 x+30$
• $y^2=55 x^6+58 x^5+44 x^4+5 x^3+38 x^2+42 x+6$
• $y^2=7 x^6+52 x^5+53 x^4+60 x^3+28 x^2+27 x+43$
• $y^2=53 x^6+4 x^5+51 x^4+35 x^3+51 x^2+4 x+53$
• $y^2=32 x^6+38 x^5+52 x^4+25 x^3+27 x^2+37 x+51$
• $y^2=15 x^6+25 x^5+35 x^4+6 x^3+49 x^2+32 x+15$
• $y^2=60 x^6+56 x^5+28 x^4+40 x^3+37 x^2+15 x+41$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.an^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
2.61.a_abv	$2$	(not in LMFDB)
2.61.ba_lf	$2$	(not in LMFDB)
2.61.ao_ff	$3$	(not in LMFDB)
2.61.ac_et	$3$	(not in LMFDB)
2.61.b_aci	$3$	(not in LMFDB)
2.61.n_ee	$3$	(not in LMFDB)
2.61.bc_mg	$3$	(not in LMFDB)