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

• Label: 2.59.ax_je
• 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:	$( 1 - 15 x + 59 x^{2} )( 1 - 8 x + 59 x^{2} )$
	$1 - 23 x + 238 x^{2} - 1357 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0692665268586$, $\pm0.325650265238$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	100
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2340$	$11934000$	$42218083440$	$146825099928000$	$511084016722550700$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3429$	$205564$	$12116921$	$714878507$	$42180019542$	$2488649320313$	$146830449570481$	$8662996070800996$	$511116755104400949$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=58 x^6+55 x^5+47 x^4+23 x^3+14 x^2+40 x+18$
• $y^2=58 x^6+50 x^5+56 x^4+11 x^3+39 x^2+18 x+58$
• $y^2=40 x^6+10 x^5+8 x^3+5 x^2+5 x+48$
• $y^2=50 x^6+21 x^5+13 x^4+40 x^3+12 x^2+18 x+47$
• $y^2=43 x^6+41 x^5+32 x^4+35 x^3+58 x^2+26 x+38$
• $y^2=43 x^6+55 x^5+57 x^4+27 x^3+53 x^2+11 x+1$
• $y^2=33 x^6+21 x^5+21 x^4+4 x^3+28 x^2+26 x+9$
• $y^2=18 x^6+47 x^5+23 x^4+47 x^3+10 x^2+43 x+45$
• $y^2=50 x^6+32 x^5+51 x^4+11 x^3+33 x^2+11 x+28$
• $y^2=44 x^6+27 x^5+44 x^4+44 x^3+29 x^2+6 x+14$
• $y^2=55 x^6+33 x^5+10 x^4+16 x^3+10 x^2+36 x+38$
• $y^2=29 x^6+46 x^5+26 x^4+33 x^3+56 x^2+29 x+5$
• $y^2=10 x^6+5 x^5+56 x^4+2 x^3+57 x^2+2 x+46$
• $y^2=52 x^6+39 x^5+50 x^4+38 x^3+23 x^2+29 x+58$
• $y^2=7 x^6+43 x^5+x^4+2 x^3+30 x^2+37 x+27$
• $y^2=34 x^6+34 x^5+28 x^4+56 x^2+6 x+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ap $\times$ 1.59.ai 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.ap : \(\Q(\sqrt{-11}) \).
• 1.59.ai : \(\Q(\sqrt{-43}) \).

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.ah_ac	$2$	(not in LMFDB)
2.59.h_ac	$2$	(not in LMFDB)
2.59.x_je	$2$	(not in LMFDB)