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

• Label: 2.59.abc_mc
• 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:	$( 1 - 14 x + 59 x^{2} )^{2}$
	$1 - 28 x + 314 x^{2} - 1652 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.135062563049$, $\pm0.135062563049$
Angle rank:	$1$ (numerical)
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2116$	$11587216$	$42071752996$	$146851740697600$	$511156771256625796$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$3326$	$204848$	$12119118$	$714980272$	$42181213646$	$2488657702528$	$146830484531998$	$8662996108797632$	$511116754593906206$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=50 x^6+57 x^5+5 x^4+7 x^3+5 x^2+57 x+50$
• $y^2=18 x^6+3 x^5+39 x^4+55 x^3+24 x^2+35 x+43$
• $y^2=53 x^6+38 x^5+10 x^4+35 x^3+21 x^2+31 x+34$
• $y^2=40 x^6+44 x^4+44 x^2+40$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

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_ada	$2$	(not in LMFDB)
2.59.bc_mc	$2$	(not in LMFDB)
2.59.o_fh	$3$	(not in LMFDB)