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

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

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$( 1 + 2 x + 59 x^{2} )^{2}$
	$1 + 4 x + 122 x^{2} + 236 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.541558382732$, $\pm0.541558382732$
Angle rank:	$1$ (numerical)
Jacobians:	$20$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3844$	$12931216$	$42038941156$	$146684265923584$	$511163200127795524$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$3710$	$204688$	$12105294$	$714989264$	$42181115726$	$2488646487776$	$146830413255454$	$8662996162178272$	$511116754050177950$

Jacobians and polarizations

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

• $y^2=5 x^5+9 x^4+22 x^2+18 x+53$
• $y^2=2 x^6+41 x^5+50 x^4+54 x^3+42 x^2+21 x+27$
• $y^2=38 x^6+14 x^5+42 x^4+12 x^3+9 x^2+44 x+39$
• $y^2=22 x^6+35 x^5+27 x^4+39 x^3+27 x^2+35 x+22$
• $y^2=54 x^6+38 x^5+8 x^4+51 x^3+48 x^2+57 x+26$
• $y^2=38 x^6+37 x^5+21 x^4+25 x^3+38 x^2+6 x+4$
• $y^2=28 x^6+25 x^5+13 x^4+13 x^3+8 x^2+37 x+48$
• $y^2=56 x^6+43 x^5+45 x^4+27 x^3+15 x^2+31 x+13$
• $y^2=25 x^6+53 x^5+52 x^4+10 x^3+43 x^2+35 x+10$
• $y^2=4 x^6+19 x^4+40 x^3+16 x^2+23 x+54$
• $y^2=8 x^5+24 x^4+54 x^3+20 x^2+28 x+40$
• $y^2=53 x^6+27 x^5+44 x^3+34 x^2+57 x+1$
• $y^2=40 x^6+13 x^5+38 x^4+45 x^3+45 x^2+18 x+2$
• $y^2=58 x^6+55 x^5+41 x^4+23 x^3+23 x^2+54 x+11$
• $y^2=18 x^6+7 x^5+55 x^4+4 x^3+51 x^2+49 x+9$
• $y^2=5 x^6+50 x^4+50 x^2+5$
• $y^2=37 x^6+22 x^5+51 x^4+40 x^3+37 x^2+29 x+12$
• $y^2=53 x^6+41 x^5+49 x^4+3 x^3+5 x^2+25 x+12$
• $y^2=51 x^6+34 x^5+45 x^4+34 x^3+35 x^2+51 x+20$
• $y^2=42 x^6+48 x^5+15 x^4+29 x^3+37 x^2+58 x+9$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.c^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	$2$	(not in LMFDB)
2.59.a_ek	$2$	(not in LMFDB)
2.59.ac_acd	$3$	(not in LMFDB)