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

• Label: 2.59.abb_ln
• 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:	$1 - 27 x + 299 x^{2} - 1593 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0994873533663$, $\pm0.201629535657$
Angle rank:	$2$ (numerical)
Number field:	4.0.46125.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2161$	$11671561$	$42130879771$	$146873114532045$	$511154665713250096$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$3351$	$205137$	$12120883$	$714977328$	$42180981531$	$2488654177287$	$146830448599123$	$8662995839363643$	$511116753302329206$

Jacobians and polarizations

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

• $y^2=39 x^6+26 x^5+39 x^4+14 x^3+51 x^2+24 x+31$
• $y^2=54 x^6+35 x^5+54 x^4+6 x^3+30 x^2+2 x+6$
• $y^2=31 x^6+8 x^5+20 x^4+20 x^3+21 x^2+31 x+55$
• $y^2=14 x^6+17 x^5+58 x^4+7 x^3+24 x^2+10 x+8$

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 4.0.46125.1.

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.bb_ln	$2$	(not in LMFDB)