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

• Label: 2.59.au_if
• 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:	$1 - 20 x + 213 x^{2} - 1180 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.206682321430$, $\pm0.331349035345$
Angle rank:	$2$ (numerical)
Number field:	4.0.379025.1
Galois group:	$D_{4}$
Jacobians:	$50$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2495$	$12213025$	$42435918080$	$146946640492025$	$511136699773927375$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$3508$	$206620$	$12126948$	$714952200$	$42180422518$	$2488650314680$	$146830437424068$	$8662995836043460$	$511116752692863348$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 50 curves (of which all are hyperelliptic):

• $y^2=39 x^6+26 x^5+45 x^4+8 x^3+42 x^2+30 x+34$
• $y^2=34 x^6+5 x^5+21 x^4+58 x^3+2 x^2+33 x+37$
• $y^2=47 x^6+19 x^5+17 x^4+26 x^3+40 x^2+27 x+11$
• $y^2=55 x^6+38 x^5+14 x^4+49 x^3+10 x^2+6 x+37$
• $y^2=57 x^6+34 x^5+46 x^4+14 x^3+36 x^2+54 x+55$
• $y^2=3 x^6+37 x^5+33 x^4+18 x^2+48 x+50$
• $y^2=18 x^6+26 x^5+36 x^4+5 x^3+48 x^2+23 x+25$
• $y^2=4 x^6+27 x^5+18 x^4+35 x^3+39 x^2+11 x+52$
• $y^2=55 x^6+47 x^5+31 x^4+45 x^3+23 x^2+21 x+38$
• $y^2=14 x^6+39 x^5+2 x^4+29 x^3+36 x^2+10 x+23$
• $y^2=12 x^6+38 x^5+24 x^4+41 x^3+41 x^2+55 x+37$
• $y^2=15 x^6+35 x^5+36 x^4+57 x^3+34 x^2+18 x+27$
• $y^2=13 x^6+x^5+17 x^4+45 x^3+9 x^2+37 x+17$
• $y^2=33 x^6+13 x^5+28 x^4+56 x^3+32 x^2+46 x+13$
• $y^2=2 x^6+13 x^5+56 x^4+57 x^3+52 x^2+14 x+30$
• $y^2=38 x^6+31 x^5+15 x^4+54 x^3+43 x^2+36 x+56$
• $y^2=33 x^6+5 x^5+32 x^4+20 x^3+5 x^2+57 x+42$
• $y^2=9 x^6+5 x^5+3 x^4+52 x^3+23 x^2+10 x+27$
• $y^2=4 x^6+37 x^5+33 x^4+20 x^3+13 x^2+53 x+42$
• $y^2=45 x^6+27 x^5+35 x^4+23 x^3+36 x^2+37 x+56$
• and

• $y^2=54 x^6+10 x^5+3 x^4+26 x^3+31 x^2+5 x+34$
• $y^2=35 x^6+53 x^5+41 x^4+29 x^3+39 x^2+56 x+47$
• $y^2=28 x^6+9 x^5+14 x^4+2 x^3+40 x^2+57 x+23$
• $y^2=30 x^6+30 x^5+27 x^4+51 x^3+35 x^2+29 x+16$
• $y^2=53 x^6+x^5+55 x^4+57 x^3+49 x^2+5 x+55$
• $y^2=31 x^6+53 x^5+46 x^4+13 x^3+50 x+49$
• $y^2=15 x^6+10 x^5+31 x^4+49 x^3+x^2+39 x+5$
• $y^2=43 x^6+25 x^5+26 x^4+12 x^3+50 x^2+46 x+52$
• $y^2=18 x^6+2 x^5+11 x^4+54 x^3+16 x^2+47 x+37$
• $y^2=2 x^6+52 x^5+45 x^4+37 x^3+43 x^2+5 x+3$
• $y^2=35 x^6+56 x^5+53 x^4+57 x^3+18 x^2+10 x+55$
• $y^2=44 x^6+18 x^5+53 x^4+53 x^3+56 x^2+11$
• $y^2=38 x^6+17 x^5+47 x^4+18 x^3+50 x^2+31 x+2$
• $y^2=58 x^6+52 x^5+38 x^4+16 x^3+7 x^2+34 x+33$
• $y^2=48 x^6+57 x^5+23 x^4+48 x^3+11 x^2+16 x+32$
• $y^2=56 x^6+x^5+36 x^4+12 x^2+20 x+37$
• $y^2=55 x^6+34 x^5+50 x^4+35 x^3+32 x^2+29 x+13$
• $y^2=33 x^6+19 x^5+41 x^4+3 x^3+5 x^2+42 x+55$
• $y^2=13 x^6+55 x^5+4 x^4+20 x^3+26 x^2+9 x+4$
• $y^2=56 x^6+26 x^5+27 x^4+36 x^3+45 x^2+4 x+41$
• $y^2=11 x^6+31 x^5+39 x^4+46 x^2+23 x+52$
• $y^2=29 x^6+49 x^5+12 x^4+29 x^3+27 x^2+40 x+33$
• $y^2=43 x^6+3 x^5+52 x^4+47 x^3+52 x^2+25 x+40$
• $y^2=2 x^6+16 x^5+36 x^4+36 x^3+x^2+9 x+16$
• $y^2=46 x^6+24 x^5+36 x^4+29 x^3+16 x^2+2 x+37$
• $y^2=10 x^6+32 x^5+46 x^4+17 x^3+32 x^2+22 x+23$
• $y^2=33 x^6+x^5+43 x^4+25 x^3+15 x^2+57 x+5$
• $y^2=16 x^6+4 x^5+36 x^4+6 x^3+58 x^2+51 x+24$
• $y^2=43 x^6+24 x^5+6 x^4+52 x^3+51 x^2+40 x+2$
• $y^2=10 x^6+46 x^5+11 x^4+42 x^3+42 x^2+36 x+37$

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 4.0.379025.1.

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.u_if	$2$	(not in LMFDB)