Abelian variety isogeny class 2.61.ax_ji over $\F_{61}$

• Label: 2.61.ax_ji
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 - 15 x + 61 x^{2} )( 1 - 8 x + 61 x^{2} )$
	$1 - 23 x + 242 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0900194921159$, $\pm0.328850104905$
Angle rank:	$2$ (numerical)
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2538$	$13679820$	$51593316768$	$191719941336000$	$713313755584881858$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$3677$	$227304$	$13846753$	$844561779$	$51519979082$	$3142742179119$	$191707341720673$	$11694146486324904$	$713342914311311477$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=31 x^6+19 x^5+8 x^4+43 x^3+48 x^2+10 x+31$
• $y^2=13 x^6+28 x^5+13 x^4+42 x^3+32 x^2+16 x+4$
• $y^2=43 x^6+27 x^5+15 x^4+49 x^3+58 x^2+56 x+6$
• $y^2=32 x^6+2 x^5+36 x^4+59 x^3+41 x^2+45 x+31$
• $y^2=35 x^6+5 x^5+36 x^4+46 x^3+18 x^2+37 x+8$
• $y^2=46 x^6+19 x^5+28 x^4+53 x^3+27 x^2+53 x+47$
• $y^2=23 x^6+8 x^5+13 x^4+32 x^3+21 x^2+35 x$
• $y^2=21 x^6+x^5+42 x^4+19 x^3+28 x^2+15 x+23$
• $y^2=40 x^6+41 x^5+58 x^4+52 x^3+49 x^2+36 x+44$
• $y^2=21 x^6+5 x^5+57 x^4+53 x^3+46 x^2+48 x+50$
• $y^2=8 x^6+51 x^5+4 x^4+32 x^3+42 x^2+32 x+26$
• $y^2=40 x^6+22 x^5+14 x^4+x^3+58 x^2+49 x+10$
• $y^2=7 x^6+23 x^5+27 x^4+24 x^3+10 x^2+40 x+59$
• $y^2=30 x^6+30 x^5+45 x^4+56 x^3+22 x^2+8 x+20$
• $y^2=12 x^6+57 x^5+27 x^4+27 x^3+3 x^2+9 x+23$
• $y^2=43 x^6+54 x^5+34 x^4+13 x^3+18 x^2+2 x+50$
• $y^2=16 x^6+60 x^5+19 x^4+43 x^3+19 x^2+31 x+2$
• $y^2=11 x^6+44 x^5+14 x^4+23 x^3+34 x^2+30 x+56$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ap $\times$ 1.61.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.61.ap : \(\Q(\sqrt{-19}) \).
• 1.61.ai : \(\Q(\sqrt{-5}) \).

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
2.61.ah_c	$2$	(not in LMFDB)
2.61.h_c	$2$	(not in LMFDB)
2.61.x_ji	$2$	(not in LMFDB)