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

• Label: 2.59.ae_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:	$\pm0.458441617268$, $\pm0.458441617268$
Angle rank:	$1$ (numerical)
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2, 29$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3364$	$12931216$	$42323187076$	$146684265923584$	$511070311443345124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$3710$	$206072$	$12105294$	$714859336$	$42181115726$	$2488656481864$	$146830413255454$	$8662995475131608$	$511116754050177950$

Jacobians and polarizations

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

• $y^2=32 x^5+34 x^4+11 x^2+9 x+56$
• $y^2=x^6+50 x^5+25 x^4+27 x^3+21 x^2+40 x+43$
• $y^2=19 x^6+7 x^5+21 x^4+6 x^3+34 x^2+22 x+49$
• $y^2=11 x^6+47 x^5+43 x^4+49 x^3+43 x^2+47 x+11$
• $y^2=27 x^6+19 x^5+4 x^4+55 x^3+24 x^2+58 x+13$
• $y^2=17 x^6+15 x^5+42 x^4+50 x^3+17 x^2+12 x+8$
• $y^2=56 x^6+50 x^5+26 x^4+26 x^3+16 x^2+15 x+37$
• $y^2=53 x^6+27 x^5+31 x^4+54 x^3+30 x^2+3 x+26$
• $y^2=50 x^6+47 x^5+45 x^4+20 x^3+27 x^2+11 x+20$
• $y^2=8 x^6+38 x^4+21 x^3+32 x^2+46 x+49$
• $y^2=4 x^5+12 x^4+27 x^3+10 x^2+14 x+20$
• $y^2=56 x^6+43 x^5+22 x^3+17 x^2+58 x+30$
• $y^2=21 x^6+26 x^5+17 x^4+31 x^3+31 x^2+36 x+4$
• $y^2=29 x^6+57 x^5+50 x^4+41 x^3+41 x^2+27 x+35$
• $y^2=9 x^6+33 x^5+57 x^4+2 x^3+55 x^2+54 x+34$
• $y^2=32 x^6+25 x^4+25 x^2+32$
• $y^2=15 x^6+44 x^5+43 x^4+21 x^3+15 x^2+58 x+24$
• $y^2=47 x^6+23 x^5+39 x^4+6 x^3+10 x^2+50 x+24$
• $y^2=55 x^6+17 x^5+52 x^4+17 x^3+47 x^2+55 x+10$
• $y^2=25 x^6+37 x^5+30 x^4+58 x^3+15 x^2+57 x+18$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ac^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.a_ek	$2$	(not in LMFDB)
2.59.e_es	$2$	(not in LMFDB)
2.59.c_acd	$3$	(not in LMFDB)