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

• Label: 2.59.ak_fn
• 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:	$( 1 - 5 x + 59 x^{2} )^{2}$
	$1 - 10 x + 143 x^{2} - 590 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.394476720982$, $\pm0.394476720982$
Angle rank:	$1$ (numerical)
Jacobians:	$30$
Cyclic group of points:	no
Non-cyclic primes:	$5, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3025$	$12780625$	$42493699600$	$146789580705625$	$511040582384550625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$3668$	$206900$	$12113988$	$714817750$	$42180199958$	$2488656102850$	$146830480381828$	$8662995760078700$	$511116750483887348$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-211}) \)$)$

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_dp	$2$	(not in LMFDB)
2.59.k_fn	$2$	(not in LMFDB)
2.59.f_abi	$3$	(not in LMFDB)