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

• Label: 2.59.b_acg
• 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:	$1 + x - 58 x^{2} + 59 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.187401536273$, $\pm0.854068202940$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-235})\)
Galois group:	$C_2^2$
Jacobians:	$22$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3484$	$11720176$	$42253269136$	$146911984233664$	$511128985948200724$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$61$	$3365$	$205732$	$12124089$	$714941411$	$42181293206$	$2488650095489$	$146830458622129$	$8662995612678268$	$511116752163579125$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{-235})\).

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

The base change of $A$ to $\F_{59^{3}}$ is 1.205379.gu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-235}) \)$)$

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.ab_acg	$2$	(not in LMFDB)
2.59.ac_ep	$3$	(not in LMFDB)
2.59.a_en	$6$	(not in LMFDB)