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

• Label: 2.59.aba_kz
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 - 26 x + 285 x^{2} - 1534 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.112413476671$, $\pm0.228039227441$
Angle rank:	$2$ (numerical)
Number field:	4.0.183872.5
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2207$	$11756689$	$42191404388$	$146898335955497$	$511158745595533447$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$3376$	$205432$	$12122964$	$714983034$	$42180912910$	$2488653019054$	$146830440000420$	$8662995821040856$	$511116753713070336$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=5 x^6+4 x^5+58 x^4+35 x^3+43 x^2+10 x+4$
• $y^2=6 x^6+50 x^5+2 x^4+7 x^3+26 x^2+55 x+11$
• $y^2=8 x^6+36 x^5+30 x^4+4 x^3+24 x^2+16 x+47$
• $y^2=48 x^6+34 x^5+35 x^4+17 x^3+8 x^2+46 x+56$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is 4.0.183872.5.

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.ba_kz	$2$	(not in LMFDB)