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

• Label: 2.59.ax_jk
• 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:	$( 1 - 14 x + 59 x^{2} )( 1 - 9 x + 59 x^{2} )$
	$1 - 23 x + 244 x^{2} - 1357 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.135062563049$, $\pm0.300760731311$
Angle rank:	$2$ (numerical)
Jacobians:	$20$
Isomorphism classes:	104

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2346$	$11978676$	$42303531816$	$146908878199200$	$511136305747439046$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3441$	$205978$	$12123833$	$714951647$	$42180537906$	$2488651609733$	$146830454021233$	$8662996076356462$	$511116755376714081$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ao $\times$ 1.59.aj 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.ao : \(\Q(\sqrt{-10}) \).
• 1.59.aj : \(\Q(\sqrt{-155}) \).

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.af_ai	$2$	(not in LMFDB)
2.59.f_ai	$2$	(not in LMFDB)
2.59.x_jk	$2$	(not in LMFDB)