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

• Label: 2.61.abc_mg
• 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:	$( 1 - 14 x + 61 x^{2} )^{2}$
	$1 - 28 x + 318 x^{2} - 1708 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.146275019398$, $\pm0.146275019398$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2304$	$13307904$	$51438240000$	$191761786404864$	$713408161597565184$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$3574$	$226618$	$13849774$	$844673554$	$51521216038$	$3142749907114$	$191707360650334$	$11694146328640258$	$713342912057332054$

Jacobians and polarizations

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

• $y^2=26x^6+53x^5+27x^4+44x^3+25x^2+40x+31$
• $y^2=25x^6+5x^4+5x^2+25$
• $y^2=38x^6+45x^5+14x^4+26x^3+14x^2+45x+38$
• $y^2=x^6+60$
• $y^2=53x^6+56x^4+56x^2+53$
• $y^2=x^6+57x^3+58$
• $y^2=55x^6+6x^5+39x^4+17x^3+39x^2+6x+55$
• $y^2=37x^5+8x^4+60x^3+8x^2+37x$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ao^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_acw	$2$	(not in LMFDB)
2.61.bc_mg	$2$	(not in LMFDB)
2.61.an_ee	$3$	(not in LMFDB)
2.61.ab_aci	$3$	(not in LMFDB)
2.61.c_et	$3$	(not in LMFDB)
2.61.o_ff	$3$	(not in LMFDB)
2.61.ba_lf	$3$	(not in LMFDB)