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

• Label: 2.59.aba_la
• 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:	$( 1 - 14 x + 59 x^{2} )( 1 - 12 x + 59 x^{2} )$
	$1 - 26 x + 286 x^{2} - 1534 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.135062563049$, $\pm0.214641822575$
Angle rank:	$2$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2208$	$11764224$	$42207538464$	$146917276139520$	$511173988737577248$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$3378$	$205510$	$12124526$	$715004354$	$42181127586$	$2488654568966$	$146830445789086$	$8662995781835170$	$511116752666045778$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=37 x^6+45 x^5+45 x^3+45 x+37$
• $y^2=14 x^6+42 x^5+40 x^4+49 x^3+38 x^2+50 x+6$
• $y^2=33 x^6+6 x^5+52 x^4+56 x^3+47 x^2+8 x+8$
• $y^2=20 x^6+24 x^5+16 x^4+9 x^3+16 x^2+24 x+20$
• $y^2=58 x^6+35 x^5+34 x^4+23 x^3+34 x^2+35 x+58$
• $y^2=8 x^6+x^5+17 x^3+x+8$
• $y^2=13 x^6+49 x^5+30 x^4+57 x^3+30 x^2+49 x+13$
• $y^2=3 x^6+31 x^5+52 x^4+45 x^3+6 x^2+30 x+16$
• $y^2=48 x^6+2 x^5+27 x^4+42 x^3+36 x^2+56 x+22$
• $y^2=53 x^6+3 x^5+20 x^4+35 x^3+20 x^2+3 x+53$
• $y^2=57 x^6+22 x^5+5 x^4+41 x^3+46 x^2+26 x+4$
• $y^2=47 x^6+22 x^5+30 x^3+22 x+47$

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

• 1.59.ao : \(\Q(\sqrt{-10}) \).
• 1.59.am : \(\Q(\sqrt{-23}) \).

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.ac_aby	$2$	(not in LMFDB)
2.59.c_aby	$2$	(not in LMFDB)
2.59.ba_la	$2$	(not in LMFDB)