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

• Label: 2.59.j_w
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 + 9 x + 22 x^{2} + 531 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.365905935355$, $\pm0.967427397978$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-155})\)
Galois group:	$C_2^2$
Jacobians:	$52$
Isomorphism classes:	72
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4044$	$11986416$	$42536587536$	$146762684362944$	$511117209422753124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$69$	$3445$	$207108$	$12111769$	$714924939$	$42179862166$	$2488654468761$	$146830444651249$	$8662996043915292$	$511116751871201125$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 52 curves (of which all are hyperelliptic):

• $y^2=56 x^6+21 x^5+12 x^4+39 x^3+39 x^2+40 x+20$
• $y^2=28 x^6+50 x^5+14 x^4+x^3+6 x^2+57 x+16$
• $y^2=43 x^6+25 x^5+53 x^4+57 x^3+42 x^2+56 x+43$
• $y^2=26 x^6+37 x^5+57 x^4+8 x^3+8 x^2+53 x+18$
• $y^2=48 x^6+49 x^5+3 x^4+40 x^3+11 x^2+40 x+1$
• $y^2=54 x^6+40 x^5+5 x^4+28 x^3+35 x^2+2 x+32$
• $y^2=33 x^6+47 x^5+33 x^4+48 x^3+57 x^2+54 x$
• $y^2=5 x^6+13 x^5+47 x^4+9 x^3+42 x^2+14 x+3$
• $y^2=9 x^6+22 x^5+7 x^4+38 x^3+37 x^2+58 x+54$
• $y^2=41 x^6+53 x^5+14 x^4+51 x^3+35 x^2+12 x+36$
• $y^2=21 x^6+42 x^5+23 x^4+56 x^3+24 x^2+29 x+45$
• $y^2=41 x^6+47 x^5+5 x^4+33 x^3+49 x^2+36 x+29$
• $y^2=22 x^6+33 x^5+58 x^4+41 x^3+29 x^2+12 x+35$
• $y^2=x^6+19 x^5+19 x^4+58 x^3+41 x^2+2 x$
• $y^2=51 x^6+29 x^5+17 x^4+17 x^3+21 x^2+53 x+35$
• $y^2=35 x^6+50 x^5+35 x^4+29 x^3+44 x^2+21 x+4$
• $y^2=42 x^6+48 x^5+53 x^4+50 x^3+30 x^2+27 x+42$
• $y^2=39 x^6+9 x^5+49 x^4+44 x^3+44 x^2+4 x+49$
• $y^2=54 x^6+57 x^5+22 x^4+9 x^3+43 x^2+53 x+26$
• $y^2=20 x^6+42 x^5+46 x^4+38 x^3+45 x^2+27 x+11$
• and

• $y^2=7 x^6+50 x^5+32 x^3+15 x^2+9 x+7$
• $y^2=24 x^6+34 x^5+44 x^4+41 x^3+4 x^2+29 x+3$
• $y^2=6 x^6+41 x^5+17 x^4+3 x^3+50 x^2+28 x+14$
• $y^2=29 x^6+12 x^4+24 x^3+13 x^2+26 x+23$
• $y^2=30 x^6+12 x^5+6 x^4+18 x^3+5 x^2+7 x+39$
• $y^2=3 x^6+9 x^5+48 x^4+28 x^3+8 x^2+32 x+30$
• $y^2=3 x^6+23 x^5+17 x^4+33 x^3+50 x^2+29 x+4$
• $y^2=56 x^6+36 x^5+23 x^4+28 x^3+38 x^2+54 x+29$
• $y^2=28 x^6+32 x^5+16 x^4+29 x^3+2 x^2+30 x$
• $y^2=26 x^6+45 x^5+19 x^4+6 x^3+39 x^2+23 x+29$
• $y^2=29 x^6+31 x^5+3 x^4+32 x^3+22 x^2+39 x+57$
• $y^2=50 x^6+57 x^5+52 x^4+9 x^3+23 x^2+28 x+12$
• $y^2=41 x^6+13 x^5+4 x^4+41 x^3+35 x^2+26 x+22$
• $y^2=45 x^6+6 x^5+6 x^4+37 x^3+36 x^2+28 x+39$
• $y^2=39 x^6+47 x^5+11 x^4+49 x^3+55 x^2+54 x+8$
• $y^2=19 x^6+45 x^5+7 x^4+38 x^3+8 x^2+10 x+19$
• $y^2=38 x^6+24 x^5+3 x^4+56 x^3+15 x^2+43 x+15$
• $y^2=14 x^6+38 x^5+35 x^4+20 x^3+55 x^2+46 x+14$
• $y^2=25 x^6+50 x^5+17 x^4+4 x^3+19 x^2+55 x+4$
• $y^2=46 x^6+51 x^5+38 x^4+48 x^3+8 x^2+22 x$
• $y^2=9 x^6+14 x^5+39 x^4+48 x^3+15 x^2+26 x+57$
• $y^2=54 x^6+47 x^5+40 x^4+44 x^3+25 x^2+57 x+47$
• $y^2=33 x^6+19 x^5+10 x^4+28 x^3+4 x^2+13 x+56$
• $y^2=5 x^6+11 x^5+2 x^4+3 x^3+41 x^2+4 x+22$
• $y^2=57 x^6+32 x^5+26 x^4+18 x^3+36 x^2+15 x+27$
• $y^2=52 x^6+10 x^5+16 x^4+2 x^3+17 x^2+36 x+56$
• $y^2=3 x^6+39 x^5+5 x^4+25 x^3+42 x^2+28 x+47$
• $y^2=7 x^6+44 x^5+10 x^4+18 x^3+56 x^2+21$
• $y^2=24 x^6+36 x^5+32 x^4+8 x^2+58 x+9$
• $y^2=4 x^6+48 x^5+55 x^4+20 x^3+12 x^2+46 x+52$
• $y^2=57 x^6+16 x^5+2 x^4+22 x^3+30 x^2+45 x+25$
• $y^2=49 x^6+27 x^5+8 x^4+36 x^3+47 x^2+44 x+8$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{3}}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-155})\).

Endomorphism algebra over $\overline{\F}_{59}$

The base change of $A$ to $\F_{59^{3}}$ is 1.205379.bhg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-155}) \)$)$

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.aj_w	$2$	(not in LMFDB)
2.59.as_hr	$3$	(not in LMFDB)
2.59.a_bl	$6$	(not in LMFDB)