Abelian variety isogeny class 2.67.e_abz over $\F_{67}$

• Label: 2.67.e_abz
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$1 + 4 x - 51 x^{2} + 268 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.245237597129$, $\pm0.911904263795$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:	$C_2^2$
Jacobians:	$28$
Isomorphism classes:	244

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4711$	$19630737$	$90904662016$	$406167349317849$	$1822931458090992151$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4372$	$302244$	$20156068$	$1350194472$	$90458490022$	$6060706742232$	$406067661717316$	$27206533871355228$	$1822837806662875732$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

• $y^2=2 x^6+4 x^3+27$
• $y^2=28 x^6+16 x^5+52 x^4+49 x^3+66 x^2+61 x+55$
• $y^2=49 x^6+64 x^5+63 x^4+34 x^3+21 x^2+23 x+36$
• $y^2=2 x^6+2 x^3+31$
• $y^2=2 x^6+35 x^5+9 x^4+46 x^3+35 x^2+x+42$
• $y^2=2 x^6+2 x^3+53$
• $y^2=30 x^6+24 x^5+14 x^4+53 x^3+66 x^2+20 x+53$
• $y^2=2 x^6+2 x^3+46$
• $y^2=2 x^6+4 x^3+52$
• $y^2=64 x^6+60 x^5+59 x^4+30 x^3+3 x^2+33 x+16$
• $y^2=37 x^6+39 x^5+54 x^4+10 x^3+56 x^2+42 x+10$
• $y^2=27 x^6+64 x^5+33 x^4+26 x^3+58 x^2+33 x+19$
• $y^2=5 x^6+7 x^5+57 x^4+38 x^3+6 x^2+2 x+50$
• $y^2=28 x^6+65 x^5+59 x^4+54 x^3+25 x^2+21 x+4$
• $y^2=37 x^6+30 x^5+51 x^4+59 x^3+17 x^2+54 x+29$
• $y^2=60 x^6+54 x^5+33 x^4+33 x^3+31 x^2+56 x+46$
• $y^2=55 x^6+46 x^5+30 x^4+39 x^3+20 x^2+2 x+48$
• $y^2=66 x^6+39 x^5+20 x^4+45 x^3+52 x^2+33 x+1$
• $y^2=29 x^6+7 x^5+55 x^4+57 x^3+49 x^2+35 x+8$
• $y^2=31 x^6+54 x^5+32 x^4+33 x^3+18 x^2+34 x+10$
• $y^2=2 x^6+2 x^3+58$
• $y^2=23 x^6+35 x^5+42 x^4+44 x^3+55 x^2+53 x+14$
• $y^2=46 x^6+36 x^5+3 x^4+51 x^3+48 x^2+51 x+40$
• $y^2=57 x^6+25 x^5+31 x^4+2 x^3+34 x^2+8 x+7$
• $y^2=2 x^6+2 x^3+12$
• $y^2=21 x^6+4 x^5+x^4+4 x^3+36 x^2+29 x+10$
• $y^2=46 x^6+7 x^5+59 x^4+11 x^3+7 x^2+43 x+24$
• $y^2=2 x^6+4 x^3+5$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67^{3}}$.

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\).

Endomorphism algebra over $\overline{\F}_{67}$

The base change of $A$ to $\F_{67^{3}}$ is 1.300763.bcm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
2.67.ae_abz	$2$	(not in LMFDB)
2.67.ai_fu	$3$	(not in LMFDB)
2.67.a_eo	$6$	(not in LMFDB)