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

• Label: 2.59.am_fu
• 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:	$( 1 - 8 x + 59 x^{2} )( 1 - 4 x + 59 x^{2} )$
	$1 - 12 x + 150 x^{2} - 708 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.325650265238$, $\pm0.416152878126$
Angle rank:	$2$ (numerical)
Jacobians:	$72$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2912$	$12673024$	$42499454816$	$146837766799360$	$511064757941689952$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$3638$	$206928$	$12117966$	$714851568$	$42180123206$	$2488652508432$	$146830457856286$	$8662995870728112$	$511116753041169878$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 72 curves (of which all are hyperelliptic):

• $y^2=13 x^6+41 x^5+15 x^4+44 x^3+37 x^2+30 x+23$
• $y^2=39 x^6+48 x^5+20 x^4+27 x^3+20 x^2+48 x+39$
• $y^2=57 x^6+45 x^5+48 x^4+15 x^3+5 x^2+21 x+4$
• $y^2=50 x^6+56 x^5+12 x^4+10 x^3+12 x^2+56 x+50$
• $y^2=9 x^6+5 x^5+35 x^4+44 x^3+44 x^2+38 x+6$
• $y^2=30 x^6+36 x^5+12 x^4+14 x^3+12 x^2+36 x+30$
• $y^2=45 x^6+32 x^5+40 x^4+14 x^3+40 x^2+32 x+45$
• $y^2=9 x^6+50 x^5+32 x^4+26 x^3+32 x^2+50 x+9$
• $y^2=56 x^6+32 x^5+54 x^4+21 x^3+54 x^2+32 x+56$
• $y^2=4 x^5+32 x^4+49 x^3+31 x^2+39 x+44$
• $y^2=8 x^6+2 x^5+25 x^4+58 x^3+25 x^2+2 x+8$
• $y^2=29 x^6+43 x^5+30 x^4+50 x^3+30 x^2+43 x+29$
• $y^2=42 x^6+24 x^5+11 x^4+39 x^3+21 x^2+20 x+26$
• $y^2=7 x^6+26 x^5+9 x^4+31 x^3+29 x^2+7 x+27$
• $y^2=8 x^6+36 x^5+40 x^4+44 x^3+40 x^2+36 x+8$
• $y^2=45 x^6+26 x^5+20 x^4+4 x^3+31 x^2+26 x+10$
• $y^2=27 x^6+34 x^5+13 x^4+20 x^3+13 x^2+34 x+27$
• $y^2=28 x^6+46 x^5+37 x^4+52 x^3+23 x^2+12 x+46$
• $y^2=31 x^6+27 x^5+25 x^4+54 x^3+25 x^2+27 x+31$
• $y^2=41 x^6+26 x^5+x^4+22 x^3+43 x^2+x+6$
• and

• $y^2=14 x^6+26 x^5+6 x^4+40 x^3+6 x^2+26 x+14$
• $y^2=26 x^6+41 x^4+12 x^3+35 x^2+33 x+21$
• $y^2=29 x^6+11 x^5+28 x^4+26 x^3+28 x^2+11 x+29$
• $y^2=6 x^6+16 x^5+10 x^4+24 x^3+5 x^2+20 x+3$
• $y^2=35 x^6+31 x^5+54 x^4+8 x^3+43 x^2+13 x+7$
• $y^2=58 x^6+12 x^5+42 x^4+22 x^3+42 x^2+12 x+58$
• $y^2=40 x^6+11 x^5+17 x^4+26 x^3+5 x^2+23 x+47$
• $y^2=35 x^6+33 x^5+39 x^4+8 x^3+40 x^2+49 x+48$
• $y^2=16 x^6+37 x^5+38 x^4+10 x^3+50 x^2+8 x+9$
• $y^2=30 x^6+55 x^5+23 x^4+48 x^2+48 x+15$
• $y^2=x^6+30 x^5+41 x^4+14 x^3+41 x^2+30 x+1$
• $y^2=39 x^6+32 x^5+25 x^4+21 x^3+25 x^2+32 x+39$
• $y^2=50 x^6+58 x^5+18 x^4+58 x^3+10 x^2+23 x+56$
• $y^2=32 x^6+42 x^4+24 x^3+19 x^2+38 x+2$
• $y^2=38 x^6+18 x^5+57 x^4+4 x^3+57 x^2+18 x+38$
• $y^2=14 x^6+42 x^5+17 x^3+42 x+14$
• $y^2=8 x^6+24 x^5+37 x^4+46 x^3+37 x^2+24 x+8$
• $y^2=37 x^6+38 x^5+16 x^4+23 x^3+38 x^2+15 x+53$
• $y^2=24 x^6+38 x^5+23 x^4+2 x^3+13 x^2+34 x+44$
• $y^2=39 x^6+46 x^5+18 x^4+52 x^3+7 x^2+15 x+14$
• $y^2=37 x^6+48 x^5+13 x^4+9 x^3+7 x^2+56 x+39$
• $y^2=47 x^6+2 x^5+48 x^4+28 x^3+8 x^2+33 x+19$
• $y^2=4 x^6+54 x^5+44 x^4+55 x^3+44 x^2+54 x+4$
• $y^2=52 x^6+11 x^5+54 x^4+8 x^3+38 x^2+43 x+2$
• $y^2=55 x^6+29 x^5+19 x^4+49 x^3+20 x^2+22 x+30$
• $y^2=31 x^6+30 x^5+6 x^4+18 x^3+32 x^2+23 x+54$
• $y^2=40 x^6+37 x^5+47 x^4+49 x^3+47 x^2+37 x+40$
• $y^2=50 x^6+14 x^5+55 x^4+35 x^3+38 x^2+54 x+39$
• $y^2=51 x^6+35 x^5+3 x^4+33 x^3+3 x^2+35 x+51$
• $y^2=27 x^6+38 x^5+53 x^4+24 x^3+19 x^2+50 x+35$
• $y^2=40 x^6+21 x^5+32 x^4+37 x^3+55 x^2+16 x+58$
• $y^2=8 x^6+32 x^5+15 x^4+42 x^3+37 x^2+33 x+11$
• $y^2=18 x^6+2 x^5+5 x^4+39 x^3+3 x^2+33 x+8$
• $y^2=27 x^6+2 x^5+52 x^4+29 x^3+56 x^2+57 x+38$
• $y^2=16 x^6+45 x^5+19 x^4+22 x^3+19 x^2+45 x+16$
• $y^2=39 x^6+28 x^5+34 x^4+46 x^3+40 x^2+9 x+16$
• $y^2=19 x^6+19 x^5+53 x^4+6 x^3+49 x^2+20 x+53$
• $y^2=40 x^6+51 x^5+27 x^4+43 x^3+38 x^2+33 x+3$
• $y^2=23 x^6+44 x^5+22 x^4+34 x^3+36 x^2+32 x+42$
• $y^2=41 x^6+37 x^5+37 x^4+29 x^3+18 x^2+19 x+4$
• $y^2=50 x^6+21 x^5+9 x^4+30 x^3+30 x^2+34 x+58$
• $y^2=13 x^5+6 x^4+50 x^3+6 x^2+13 x$
• $y^2=19 x^6+34 x^5+3 x^4+28 x^3+26 x^2+43 x+22$
• $y^2=46 x^6+6 x^5+27 x^4+38 x^3+40 x^2+12 x+52$
• $y^2=23 x^6+50 x^5+50 x^4+16 x^3+50 x^2+50 x+23$
• $y^2=32 x^6+34 x^5+30 x^4+14 x^3+34 x^2+40 x+23$
• $y^2=36 x^6+46 x^5+18 x^4+27 x^3+18 x^2+46 x+36$
• $y^2=8 x^6+4 x^5+8 x^4+29 x^3+8 x^2+4 x+8$
• $y^2=58 x^6+46 x^5+10 x^4+20 x^3+10 x^2+46 x+58$
• $y^2=50 x^6+20 x^5+39 x^4+42 x^3+18 x^2+28 x+55$
• $y^2=30 x^6+43 x^5+13 x^4+46 x^3+30 x^2+30 x+54$
• $y^2=23 x^6+11 x^5+44 x^4+11 x^3+24 x^2+14 x+54$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ai $\times$ 1.59.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.59.ai : \(\Q(\sqrt{-43}) \).
• 1.59.ae : \(\Q(\sqrt{-55}) \).

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.ae_di	$2$	(not in LMFDB)
2.59.e_di	$2$	(not in LMFDB)
2.59.m_fu	$2$	(not in LMFDB)