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

• Label: 2.59.ak_fm
• 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:	$( 1 - 6 x + 59 x^{2} )( 1 - 4 x + 59 x^{2} )$
	$1 - 10 x + 142 x^{2} - 590 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.372279067924$, $\pm0.416152878126$
Angle rank:	$2$ (numerical)
Jacobians:	$56$
Isomorphism classes:	224
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3024$	$12773376$	$42487505424$	$146791636992000$	$511045086079244304$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$3666$	$206870$	$12114158$	$714824050$	$42180224706$	$2488655553910$	$146830474169758$	$8662995773196530$	$511116751129934706$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 56 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ag $\times$ 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.ag : \(\Q(\sqrt{-2}) \).
• 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.ac_dq	$2$	(not in LMFDB)
2.59.c_dq	$2$	(not in LMFDB)
2.59.k_fm	$2$	(not in LMFDB)