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

• Label: 2.59.p_fc
• 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:	$( 1 + x + 59 x^{2} )( 1 + 14 x + 59 x^{2} )$
	$1 + 15 x + 132 x^{2} + 885 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.520734869606$, $\pm0.864937436951$
Angle rank:	$2$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4514$	$12250996$	$42199381784$	$146759581482400$	$511108979449859054$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$75$	$3521$	$205470$	$12111513$	$714913425$	$42181253426$	$2488646986635$	$146830440050353$	$8662995776571930$	$511116755084336081$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=9 x^6+17 x^5+23 x^4+30 x^3+53 x^2+8 x+46$
• $y^2=3 x^6+45 x^5+15 x^4+28 x^3+29 x^2+47 x+24$
• $y^2=14 x^6+48 x^5+5 x^4+32 x^3+41 x^2+15 x+24$
• $y^2=24 x^6+35 x^5+12 x^4+40 x^3+44 x^2+26 x+16$
• $y^2=12 x^6+43 x^5+8 x^4+46 x^3+43 x^2+41 x+23$
• $y^2=38 x^6+26 x^5+26 x^4+23 x^3+38 x^2+54 x+21$
• $y^2=28 x^6+38 x^5+27 x^4+33 x^3+54 x^2+18 x+17$
• $y^2=19 x^6+23 x^4+38 x^3+56 x^2+8 x$
• $y^2=16 x^6+31 x^5+37 x^4+19 x^3+2 x^2+29 x+7$
• $y^2=29 x^6+49 x^5+56 x^4+36 x^3+4 x^2+44 x+2$
• $y^2=9 x^6+4 x^5+2 x^4+16 x^3+57 x^2+51 x+39$
• $y^2=19 x^6+16 x^5+56 x^4+14 x^3+52 x^2+16 x+51$
• $y^2=28 x^6+57 x^5+25 x^4+13 x^3+50 x^2+42 x+55$
• $y^2=36 x^6+13 x^5+14 x^4+38 x^3+54 x^2+19 x+29$
• $y^2=44 x^6+8 x^5+25 x^4+53 x^3+19 x^2+37 x+12$
• $y^2=20 x^6+6 x^5+45 x^4+5 x^3+42 x^2+33 x+45$
• $y^2=43 x^5+11 x^4+55 x^3+23 x^2+21 x+1$
• $y^2=20 x^6+26 x^5+29 x^4+54 x^3+46 x^2+26 x+8$
• $y^2=38 x^6+22 x^5+24 x^4+14 x^3+36 x^2+27 x+7$
• $y^2=41 x^6+11 x^5+25 x^4+x^3+36 x^2+27 x+6$
• $y^2=53 x^6+38 x^5+43 x^4+11 x^3+39 x^2+20 x+11$
• $y^2=32 x^6+38 x^5+27 x^4+14 x^3+9 x^2+32 x+25$
• $y^2=15 x^6+26 x^5+39 x^4+22 x^3+57 x^2+20 x+17$
• $y^2=48 x^6+23 x^5+24 x^4+57 x^3+43 x^2+36 x+41$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.b $\times$ 1.59.o 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.b : \(\Q(\sqrt{-235}) \).
• 1.59.o : \(\Q(\sqrt{-10}) \).

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.ap_fc	$2$	(not in LMFDB)
2.59.an_ea	$2$	(not in LMFDB)
2.59.n_ea	$2$	(not in LMFDB)