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

• Label: 2.61.ay_jx
• 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:	$( 1 - 15 x + 61 x^{2} )( 1 - 9 x + 61 x^{2} )$
	$1 - 24 x + 257 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0900194921159$, $\pm0.304548188780$
Angle rank:	$2$ (numerical)
Jacobians:	$11$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2491$	$13618297$	$51585620800$	$191743238558025$	$713331744448229731$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3660$	$227270$	$13848436$	$844583078$	$51520042662$	$3142740999758$	$191707325161636$	$11694146420296190$	$713342914941697980$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):

• $y^2=4 x^6+48 x^5+39 x^3+48 x+4$
• $y^2=8 x^6+36 x^5+60 x^4+9 x^3+48 x^2+59 x+59$
• $y^2=4 x^6+51 x^5+36 x^4+44 x^3+33 x^2+48 x+38$
• $y^2=47 x^6+3 x^5+60 x^4+44 x^3+52 x^2+15 x+31$
• $y^2=42 x^6+54 x^5+51 x^4+55 x^3+50 x^2+11 x+21$
• $y^2=53 x^6+10 x^5+x^4+15 x^3+34 x^2+7 x+31$
• $y^2=40 x^6+53 x^5+x^4+17 x^3+46 x^2+27 x+30$
• $y^2=x^6+33 x^5+12 x^4+59 x^3+6 x^2+2 x+26$
• $y^2=11 x^6+38 x^5+59 x^4+27 x^3+59 x^2+38 x+11$
• $y^2=21 x^6+26 x^5+52 x^4+5 x^3+52 x^2+26 x+21$
• $y^2=5 x^6+6 x^5+17 x^4+10 x^3+53 x^2+39 x+55$

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 $\times$ 1.61.aj 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.ap : \(\Q(\sqrt{-19}) \).
• 1.61.aj : \(\Q(\sqrt{-163}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.61.ag_an	$2$	(not in LMFDB)
2.61.g_an	$2$	(not in LMFDB)
2.61.y_jx	$2$	(not in LMFDB)