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

• Label: 2.59.ax_jm
• 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:	$1 - 23 x + 246 x^{2} - 1357 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.155667863679$, $\pm0.289401932737$
Angle rank:	$2$ (numerical)
Number field:	4.0.552857.1
Galois group:	$D_{4}$
Jacobians:	$12$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2348$	$11993584$	$42332026256$	$146936419967168$	$511152420522585988$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3445$	$206116$	$12126105$	$714974187$	$42180657814$	$2488651370809$	$146830441698033$	$8662995935701756$	$511116754398709605$

Jacobians and polarizations

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

• $y^2=41 x^6+20 x^5+14 x^4+55 x^3+56 x^2+50 x+14$
• $y^2=32 x^6+57 x^5+43 x^4+52 x^3+32 x^2+52 x+8$
• $y^2=14 x^6+55 x^5+39 x^4+42 x^3+58 x^2+39 x+5$
• $y^2=48 x^6+55 x^5+50 x^4+32 x^3+12 x^2+26 x+44$
• $y^2=37 x^6+3 x^5+16 x^4+16 x^3+23 x^2+36 x+30$
• $y^2=9 x^6+x^5+7 x^4+22 x^3+14 x^2+46 x+53$
• $y^2=38 x^6+11 x^5+56 x^4+25 x^3+16 x^2+26 x+29$
• $y^2=2 x^6+27 x^5+24 x^4+54 x^3+43 x^2+51 x+37$
• $y^2=9 x^6+3 x^5+14 x^4+9 x^3+47 x^2+14 x+58$
• $y^2=52 x^6+7 x^5+40 x^4+54 x^3+15 x^2+25 x+2$
• $y^2=50 x^6+15 x^5+x^4+44 x^3+13 x^2+53 x+12$
• $y^2=9 x^6+13 x^5+11 x^4+44 x^3+8 x^2+10 x+52$

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 4.0.552857.1.

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_jm	$2$	(not in LMFDB)