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

• Label: 2.59.k_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:	$\pm0.392293639867$, $\pm0.941039693466$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4123$	$12051529$	$42497822500$	$146750034501049$	$511101775360534603$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$3464$	$206920$	$12110724$	$714903350$	$42180169358$	$2488654542290$	$146830457432644$	$8662995782869960$	$511116752309695304$

Jacobians and polarizations

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

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

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

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.bdq^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.ak_bp	$2$	(not in LMFDB)
2.59.au_ik	$3$	(not in LMFDB)
2.59.a_s	$6$	(not in LMFDB)