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

• Label: 2.59.ay_jv
• 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:	$1 - 24 x + 255 x^{2} - 1416 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0976023068366$, $\pm0.291607162987$
Angle rank:	$2$ (numerical)
Number field:	4.0.2503312.2
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2297$	$11891569$	$42239634068$	$146875063791817$	$511120340776624097$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3416$	$205668$	$12121044$	$714929316$	$42180352670$	$2488650073596$	$146830444168420$	$8662996062338364$	$511116756106868936$

Jacobians and polarizations

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

• $y^2=14 x^6+57 x^5+21 x^4+3 x^3+47 x^2+39 x+14$
• $y^2=14 x^6+53 x^5+25 x^4+51 x^3+58 x^2+46 x+32$
• $y^2=37 x^6+43 x^5+33 x^4+57 x^3+47 x^2+20$
• $y^2=23 x^6+14 x^5+32 x^4+9 x^3+23 x^2+27 x+37$

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 4.0.2503312.2.

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.y_jv	$2$	(not in LMFDB)