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

• Label: 2.59.az_km
• 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:	$( 1 - 14 x + 59 x^{2} )( 1 - 11 x + 59 x^{2} )$
	$1 - 25 x + 272 x^{2} - 1475 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.135062563049$, $\pm0.245953251861$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2254$	$11842516$	$42252663544$	$146925359005600$	$511165460071757554$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$3401$	$205730$	$12125193$	$714992425$	$42180904946$	$2488652569795$	$146830436958673$	$8662995817930790$	$511116753765983081$

Jacobians and polarizations

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

• $y^2=32 x^6+58 x^5+15 x^4+58 x^3+24 x^2+25 x+14$
• $y^2=45 x^6+52 x^5+16 x^4+35 x^3+5 x^2+58 x$
• $y^2=14 x^6+17 x^5+30 x^4+2 x^3+16 x^2+53 x+28$
• $y^2=11 x^6+5 x^5+2 x^4+50 x^3+2 x^2+38 x+33$
• $y^2=42 x^6+31 x^5+45 x^4+5 x^3+50 x^2+26 x+1$
• $y^2=27 x^6+4 x^5+5 x^4+32 x^3+51 x^2+33 x+55$
• $y^2=55 x^6+45 x^5+58 x^4+57 x^3+44 x^2+45 x+52$
• $y^2=39 x^6+30 x^5+27 x^4+27 x^3+26 x^2+22 x+49$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ao $\times$ 1.59.al 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.ao : \(\Q(\sqrt{-10}) \).
• 1.59.al : \(\Q(\sqrt{-115}) \).

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.ad_abk	$2$	(not in LMFDB)
2.59.d_abk	$2$	(not in LMFDB)
2.59.z_km	$2$	(not in LMFDB)