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

• Label: 2.59.a_aek
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 - 114 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.0415583827317$, $\pm0.958441617268$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{58})\)
Galois group:	$C_2^2$
Jacobians:	$3$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3368$	$11343424$	$42180242600$	$146684265923584$	$511116752925873128$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3254$	$205380$	$12105294$	$714924300$	$42179951558$	$2488651484820$	$146830413255454$	$8662995818654940$	$511116752551104854$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=2 x^5+40 x^4+29 x^3+14 x^2+2 x+18$
• $y^2=8 x^6+36 x^5+22 x^4+7 x^2+30 x+39$
• $y^2=6 x^6+42 x^5+31 x^4+41 x^3+55 x^2+55 x+40$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{2}}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{58})\).

Endomorphism algebra over $\overline{\F}_{59}$

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.aek^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-58}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.59.ae_es	$4$	(not in LMFDB)
2.59.a_ek	$4$	(not in LMFDB)
2.59.e_es	$4$	(not in LMFDB)