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

• Label: 2.59.a_en
• 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:	$( 1 - x + 59 x^{2} )( 1 + x + 59 x^{2} )$
	$1 + 117 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.479265130394$, $\pm0.520734869606$
Angle rank:	$1$ (numerical)
Jacobians:	$10$
Isomorphism classes:	16
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3599$	$12952801$	$42180913424$	$146667480103225$	$511116754437703679$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3716$	$205380$	$12103908$	$714924300$	$42181293206$	$2488651484820$	$146830395568708$	$8662995818654940$	$511116755574765956$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=51 x^6+5 x^5+19 x^4+8 x^3+48 x^2+19 x+53$
• $y^2=43 x^6+10 x^5+38 x^4+16 x^3+37 x^2+38 x+47$
• $y^2=29 x^6+56 x^5+52 x^4+6 x^3+33 x^2+14 x+48$
• $y^2=58 x^6+53 x^5+45 x^4+12 x^3+7 x^2+28 x+37$
• $y^2=5 x^6+58 x^5+54 x^4+10 x^3+50 x^2+18 x+15$
• $y^2=10 x^6+57 x^5+49 x^4+20 x^3+41 x^2+36 x+30$
• $y^2=50 x^6+29 x^5+44 x^4+57 x^3+38 x^2+12 x+31$
• $y^2=41 x^6+58 x^5+29 x^4+55 x^3+17 x^2+24 x+3$
• $y^2=43 x^6+42 x^5+16 x^4+13 x^3+21 x^2+47 x+6$
• $y^2=27 x^6+25 x^5+32 x^4+26 x^3+42 x^2+35 x+12$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{2}}$.

Endomorphism algebra over $\F_{59}$

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

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

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.en^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-235}) \)$)$

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.ac_ep	$2$	(not in LMFDB)
2.59.c_ep	$2$	(not in LMFDB)
2.59.a_aen	$4$	(not in LMFDB)