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

• Label: 2.59.ax_jl
• 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:	$1 - 23 x + 245 x^{2} - 1357 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.145210552306$, $\pm0.295391543398$
Angle rank:	$2$ (numerical)
Number field:	4.0.122157.1
Galois group:	$D_{4}$
Jacobians:	$10$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2347$	$11986129$	$42317778301$	$146922673096525$	$511144445313323152$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3443$	$206047$	$12124971$	$714963032$	$42180601151$	$2488651551773$	$146830448676931$	$8662996013705083$	$511116754929861878$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=53 x^6+20 x^5+58 x^4+49 x^3+17 x^2+27 x+52$
• $y^2=18 x^6+34 x^5+50 x^4+x^3+31 x^2+27 x+11$
• $y^2=16 x^6+25 x^5+13 x^4+27 x^3+41 x^2+x+58$
• $y^2=34 x^6+23 x^5+13 x^4+43 x^3+26 x^2+4 x+27$
• $y^2=32 x^6+58 x^5+41 x^4+21 x^3+47 x^2+2 x+32$
• $y^2=31 x^6+39 x^5+53 x^4+43 x^3+3 x^2+3 x+23$
• $y^2=2 x^6+26 x^5+48 x^4+6 x^3+58 x^2+34 x+43$
• $y^2=6 x^6+4 x^5+42 x^4+12 x^3+3 x^2+46 x+49$
• $y^2=2 x^6+23 x^5+12 x^4+19 x^3+54 x^2+21 x+42$
• $y^2=54 x^6+23 x^5+25 x^4+10 x^3+9 x^2+2 x+39$

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 4.0.122157.1.

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.x_jl	$2$	(not in LMFDB)