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

• Label: 2.59.ak_bp
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 - 10 x + 41 x^{2} - 590 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0589603065338$, $\pm0.607706360133$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-34})\)
Galois group:	$C_2^2$
Jacobians:	$56$
Cyclic group of points:	yes

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2923$	$12051529$	$41865252100$	$146750034501049$	$511131730688704603$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$3464$	$203840$	$12110724$	$714945250$	$42180169358$	$2488648427350$	$146830457432644$	$8662995854439920$	$511116752309695304$

Jacobians and polarizations

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

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{3}}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-34})\).

Endomorphism algebra over $\overline{\F}_{59}$

The base change of $A$ to $\F_{59^{3}}$ is 1.205379.abdq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-34}) \)$)$

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.k_bp	$2$	(not in LMFDB)
2.59.u_ik	$3$	(not in LMFDB)
2.59.au_ik	$6$	(not in LMFDB)