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

• Label: 2.59.az_kh
• 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:	$1 - 25 x + 267 x^{2} - 1475 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0473576671055$, $\pm0.279593066244$
Angle rank:	$2$ (numerical)
Number field:	4.0.609725.2
Galois group:	$C_4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2249$	$11805001$	$42175161899$	$146839161063725$	$511100668270704304$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$3391$	$205355$	$12118083$	$714901800$	$42180058531$	$2488646765045$	$146830410767283$	$8662995785442665$	$511116754328017206$

Jacobians and polarizations

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

• $y^2=50 x^6+10 x^5+38 x^4+43 x^3+14 x^2+44 x+18$
• $y^2=36 x^6+43 x^5+15 x^4+35 x^3+14 x^2+53 x+18$
• $y^2=34 x^6+58 x^5+27 x^4+45 x^3+50 x^2+17 x+39$
• $y^2=47 x^6+6 x^5+32 x^4+11 x^3+39 x^2+26 x+4$

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 4.0.609725.2.

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.z_kh	$2$	(not in LMFDB)