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

• Label: 2.59.bc_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:	$\pm0.864937436951$, $\pm0.864937436951$
Angle rank:	$1$ (numerical)
Jacobians:	$4$
Cyclic group of points:	no
Non-cyclic primes:	$2, 37$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5476$	$11587216$	$42290277316$	$146851740697600$	$511076739770786596$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$88$	$3326$	$205912$	$12119118$	$714868328$	$42181213646$	$2488645267112$	$146830484531998$	$8662995528512248$	$511116754593906206$

Jacobians and polarizations

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

• $y^2=25 x^6+58 x^5+32 x^4+33 x^3+32 x^2+58 x+25$
• $y^2=9 x^6+31 x^5+49 x^4+57 x^3+12 x^2+47 x+51$
• $y^2=47 x^6+17 x^5+20 x^4+11 x^3+42 x^2+3 x+9$
• $y^2=21 x^6+29 x^4+29 x^2+21$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.o^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.abc_mc	$2$	(not in LMFDB)
2.59.a_ada	$2$	(not in LMFDB)
2.59.ao_fh	$3$	(not in LMFDB)