Abelian variety isogeny class 2.59.az_ki over $\F_{59}$

• Label: 2.59.az_ki
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$( 1 - 15 x + 59 x^{2} )( 1 - 10 x + 59 x^{2} )$
	$1 - 25 x + 268 x^{2} - 1475 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0692665268586$, $\pm0.274373026800$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	52

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2250$	$11812500$	$42190659000$	$146856496500000$	$511113983878743750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$3393$	$205430$	$12119513$	$714920425$	$42180243858$	$2488648277395$	$146830421877553$	$8662995871133090$	$511116755105155473$

Jacobians and polarizations

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

• $y^2=13 x^6+54 x^5+23 x^4+22 x^3+20 x^2+13 x+32$
• $y^2=11 x^6+45 x^5+13 x^4+58 x^3+38 x^2+57 x+24$
• $y^2=56 x^6+7 x^5+37 x^4+37 x^3+53 x^2+36 x+31$
• $y^2=2 x^6+8 x^5+41 x^4+33 x^3+32 x^2+40 x+33$
• $y^2=24 x^6+26 x^5+51 x^4+16 x^3+5 x^2+52 x+15$
• $y^2=43 x^6+35 x^5+54 x^3+8 x^2+14 x+37$
• $y^2=39 x^6+11 x^5+25 x^4+29 x^3+14 x^2+19 x+4$
• $y^2=25 x^6+39 x^5+57 x^4+36 x^3+25 x^2+6 x+50$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59}$.

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ap $\times$ 1.59.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.59.ap : \(\Q(\sqrt{-11}) \).
• 1.59.ak : \(\Q(\sqrt{-34}) \).

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
2.59.af_abg	$2$	(not in LMFDB)
2.59.f_abg	$2$	(not in LMFDB)
2.59.z_ki	$2$	(not in LMFDB)