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

• Label: 2.59.ag_ev
• 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:	$1 - 6 x + 125 x^{2} - 354 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.407228766241$, $\pm0.467083539198$
Angle rank:	$2$ (numerical)
Number field:	4.0.3235392.4
Galois group:	$D_{4}$
Jacobians:	$20$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3247$	$12880849$	$42380766148$	$146719993219497$	$511059872468133847$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$3696$	$206352$	$12108244$	$714844734$	$42180795150$	$2488656388314$	$146830437910180$	$8662995577096416$	$511116752638585536$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=39 x^6+30 x^5+43 x^4+24 x^3+35 x^2+44 x+26$
• $y^2=13 x^6+41 x^5+37 x^4+45 x^3+54 x^2+24 x+18$
• $y^2=50 x^6+29 x^5+28 x^4+34 x^3+12 x^2+21 x+35$
• $y^2=17 x^6+56 x^5+32 x^4+34 x^3+51 x^2+37 x+56$
• $y^2=40 x^6+38 x^5+37 x^4+17 x^3+8 x^2+20 x+35$
• $y^2=42 x^6+47 x^5+50 x^4+28 x^3+46 x^2+55 x+29$
• $y^2=32 x^6+6 x^5+3 x^4+18 x^3+3 x^2+14 x+2$
• $y^2=24 x^6+23 x^5+12 x^4+29 x^3+37 x^2+36 x+50$
• $y^2=32 x^6+47 x^5+41 x^4+19 x^3+4 x^2+3 x+26$
• $y^2=21 x^6+33 x^5+32 x^4+15 x^3+13 x^2+37 x+58$
• $y^2=15 x^6+5 x^5+32 x^4+49 x^3+58 x^2+49 x+35$
• $y^2=2 x^6+28 x^5+6 x^4+43 x^3+51 x^2+47 x+11$
• $y^2=37 x^6+11 x^5+34 x^4+20 x^3+13 x^2+3 x+45$
• $y^2=7 x^6+6 x^5+55 x^4+52 x^3+35 x^2+17 x+42$
• $y^2=10 x^6+35 x^5+54 x^4+39 x^3+26 x^2+54 x+33$
• $y^2=26 x^6+37 x^5+32 x^4+15 x^3+52 x^2+28 x+4$
• $y^2=28 x^6+42 x^5+8 x^4+6 x^3+33 x^2+53 x+28$
• $y^2=50 x^6+53 x^5+58 x^4+17 x^3+50 x^2+7 x+12$
• $y^2=54 x^6+43 x^5+23 x^4+35 x^3+39 x^2+26 x+45$
• $y^2=27 x^6+48 x^5+48 x^4+29 x^3+54 x^2+26 x+10$

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 4.0.3235392.4.

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.g_ev	$2$	(not in LMFDB)