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

• Label: 2.61.aba_le
• 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:	$( 1 - 14 x + 61 x^{2} )( 1 - 12 x + 61 x^{2} )$
	$1 - 26 x + 290 x^{2} - 1586 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.146275019398$, $\pm0.221142061624$
Angle rank:	$2$ (numerical)
Jacobians:	$10$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2400$	$13497600$	$51585660000$	$191830914662400$	$713421947728860000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3626$	$227268$	$13854766$	$844689876$	$51521030138$	$3142745838996$	$191707316101726$	$11694145994559108$	$713342910530017226$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=30 x^6+10 x^5+51 x^4+13 x^3+51 x^2+10 x+30$
• $y^2=46 x^6+47 x^4+14 x^3+12 x^2+5$
• $y^2=11 x^6+49 x^5+52 x^4+37 x^3+52 x^2+49 x+11$
• $y^2=26 x^6+4 x^5+41 x^4+22 x^3+41 x^2+4 x+26$
• $y^2=14 x^6+23 x^5+11 x^4+47 x^3+8 x^2+53 x+36$
• $y^2=51 x^6+24 x^5+39 x^4+24 x^3+39 x^2+24 x+51$
• $y^2=33 x^6+6 x^5+7 x^4+21 x^3+6 x^2+43 x+11$
• $y^2=5 x^6+25 x^5+58 x^4+25 x^3+22 x^2+16 x+5$
• $y^2=4 x^6+21 x^5+33 x^4+20 x^3+43 x^2+37 x+12$
• $y^2=44 x^6+52 x^5+60 x^4+31 x^3+60 x^2+52 x+44$

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 $\times$ 1.61.am and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.61.ao : \(\Q(\sqrt{-3}) \).
• 1.61.am : \(\Q(\sqrt{-1}) \).

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
2.61.ac_abu	$2$	(not in LMFDB)
2.61.c_abu	$2$	(not in LMFDB)
2.61.ba_le	$2$	(not in LMFDB)
2.61.al_eg	$3$	(not in LMFDB)
2.61.b_abi	$3$	(not in LMFDB)