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

• Label: 2.59.n_gd
• 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:	$1 + 13 x + 159 x^{2} + 767 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.613932648544$, $\pm0.665158734200$
Angle rank:	$2$ (numerical)
Number field:	4.0.921125.1
Galois group:	$D_{4}$
Jacobians:	$14$
Isomorphism classes:	14
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4421$	$12648481$	$41831904311$	$146859552376445$	$511173970144978096$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$73$	$3631$	$203677$	$12119763$	$715004328$	$42179899051$	$2488651096627$	$146830473811443$	$8662995617838883$	$511116752661073206$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=12 x^6+51 x^5+5 x^4+15 x^3+27 x^2+20 x+7$
• $y^2=x^6+22 x^5+16 x^4+58 x^3+43 x^2+28 x+8$
• $y^2=43 x^6+54 x^5+9 x^4+21 x^3+16 x^2+25 x+26$
• $y^2=29 x^6+42 x^5+13 x^4+10 x^3+30 x^2+22 x+49$
• $y^2=50 x^6+36 x^5+46 x^4+58 x^3+28 x^2+21 x+48$
• $y^2=20 x^6+52 x^5+29 x^4+33 x^3+29 x^2+5 x+11$
• $y^2=57 x^6+28 x^5+19 x^4+32 x^3+8 x^2+46 x+31$
• $y^2=57 x^6+52 x^5+19 x^4+7 x^3+47 x^2+18 x+25$
• $y^2=12 x^6+5 x^5+12 x^4+49 x^3+12 x^2+3 x+46$
• $y^2=23 x^6+19 x^5+26 x^4+7 x^3+13 x^2+42 x+48$
• $y^2=36 x^6+55 x^5+28 x^4+21 x^3+21 x^2+41 x+37$
• $y^2=18 x^6+14 x^5+55 x^4+8 x^3+16 x^2+10 x+25$
• $y^2=53 x^6+49 x^5+30 x^4+47 x^3+21 x^2+2 x+3$
• $y^2=25 x^6+14 x^5+54 x^4+20 x^3+20 x^2+47 x+57$

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 4.0.921125.1.

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.an_gd	$2$	(not in LMFDB)