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

• Label: 2.59.o_gk
• 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:	$( 1 + 6 x + 59 x^{2} )( 1 + 8 x + 59 x^{2} )$
	$1 + 14 x + 166 x^{2} + 826 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.627720932076$, $\pm0.674349734762$
Angle rank:	$2$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4488$	$12602304$	$41822294184$	$146882373580800$	$511166410099476648$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$3618$	$203630$	$12121646$	$714993754$	$42179822226$	$2488652468446$	$146830469647006$	$8662995552635690$	$511116753369789378$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=3 x^6+24 x^5+56 x^4+50 x^3+56 x^2+24 x+3$
• $y^2=48 x^6+32 x^5+45 x^4+5 x^3+45 x^2+32 x+48$
• $y^2=51 x^6+3 x^5+45 x^4+31 x^3+25 x^2+49 x+17$
• $y^2=49 x^6+15 x^5+13 x^4+52 x^3+13 x^2+15 x+49$
• $y^2=47 x^6+38 x^5+21 x^4+16 x^3+21 x^2+38 x+47$
• $y^2=56 x^6+29 x^5+5 x^4+5 x^3+17 x^2+19 x+10$
• $y^2=49 x^6+52 x^5+35 x^4+51 x^3+35 x^2+52 x+49$
• $y^2=57 x^6+16 x^5+10 x^4+30 x^3+10 x^2+16 x+57$
• $y^2=27 x^5+32 x^4+15 x^3+50 x^2+3 x$
• $y^2=55 x^6+45 x^5+31 x^4+9 x^3+31 x^2+45 x+55$
• $y^2=25 x^6+10 x^5+16 x^4+14 x^3+16 x^2+10 x+25$
• $y^2=54 x^6+31 x^5+26 x^4+57 x^3+26 x^2+31 x+54$
• $y^2=7 x^6+53 x^5+5 x^4+39 x^3+22 x^2+16 x+28$
• $y^2=53 x^6+37 x^5+37 x^4+41 x^3+10 x^2+33 x+12$
• $y^2=51 x^6+25 x^5+22 x^4+13 x^3+41 x^2+26 x+48$
• $y^2=48 x^6+20 x^5+32 x^4+2 x^3+52 x^2+27 x+5$
• $y^2=50 x^6+51 x^5+10 x^3+51 x+50$
• $y^2=32 x^6+51 x^5+48 x^4+26 x^3+48 x^2+51 x+32$
• $y^2=2 x^6+42 x^5+49 x^4+41 x^3+20 x^2+50 x+43$
• $y^2=44 x^6+8 x^5+22 x^4+x^3+26 x^2+18 x+58$
• and

• $y^2=19 x^6+25 x^5+23 x^4+41 x^3+23 x^2+25 x+19$
• $y^2=3 x^6+58 x^5+19 x^4+15 x^3+19 x^2+58 x+3$
• $y^2=56 x^6+34 x^5+3 x^4+25 x^3+17 x^2+56 x+31$
• $y^2=32 x^6+33 x^4+37 x^3+33 x^2+32$
• $y^2=34 x^6+17 x^5+47 x^3+17 x+34$
• $y^2=34 x^6+43 x^5+8 x^4+58 x^3+43 x^2+54 x+23$
• $y^2=5 x^6+28 x^5+24 x^4+48 x^3+56 x^2+41 x+22$
• $y^2=56 x^6+33 x^5+22 x^4+16 x^3+22 x^2+33 x+56$
• $y^2=33 x^6+x^5+12 x^4+19 x^3+41 x^2+17 x+14$
• $y^2=30 x^6+58 x^5+43 x^4+50 x^3+43 x^2+58 x+30$
• $y^2=55 x^6+2 x^5+54 x^4+34 x^3+32 x^2+30 x+30$
• $y^2=22 x^6+13 x^5+14 x^4+x^3+14 x^2+13 x+22$
• $y^2=45 x^6+34 x^5+22 x^4+41 x^3+25 x^2+50 x+5$
• $y^2=46 x^6+13 x^5+43 x^4+42 x^3+43 x^2+13 x+46$
• $y^2=20 x^6+5 x^5+34 x^4+9 x^3+34 x^2+5 x+20$
• $y^2=28 x^6+49 x^5+9 x^4+41 x^2+19 x+12$
• $y^2=x^6+28 x^5+25 x^4+32 x^3+25 x^2+28 x+1$
• $y^2=11 x^6+19 x^5+16 x^4+24 x^3+35 x^2+28 x+44$
• $y^2=17 x^6+54 x^5+3 x^4+51 x^3+22 x^2+13 x+22$
• $y^2=35 x^6+15 x^5+10 x^4+47 x^3+10 x^2+15 x+35$
• $y^2=51 x^6+27 x^5+41 x^4+51 x^3+9 x^2+51 x+1$
• $y^2=31 x^6+13 x^5+24 x^4+13 x^3+24 x^2+13 x+31$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.g $\times$ 1.59.i 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.g : \(\Q(\sqrt{-2}) \).
• 1.59.i : \(\Q(\sqrt{-43}) \).

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.ao_gk	$2$	(not in LMFDB)
2.59.ac_cs	$2$	(not in LMFDB)
2.59.c_cs	$2$	(not in LMFDB)