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

• Label: 2.67.ae_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:	$\pm0.0880957362045$, $\pm0.754762402871$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:	$C_2^2$
Jacobians:	$28$
Isomorphism classes:	244
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4167$	$19630737$	$90014400576$	$406167349317849$	$1822744157935009527$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$4372$	$299284$	$20156068$	$1350055744$	$90458490022$	$6060716468416$	$406067661717316$	$27206534921234668$	$1822837806662875732$

Jacobians and polarizations

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

• $y^2=x^6+2 x^3+47$
• $y^2=56 x^6+32 x^5+37 x^4+31 x^3+65 x^2+55 x+43$
• $y^2=31 x^6+61 x^5+59 x^4+x^3+42 x^2+46 x+5$
• $y^2=x^6+x^3+49$
• $y^2=4 x^6+3 x^5+18 x^4+25 x^3+3 x^2+2 x+17$
• $y^2=x^6+x^3+60$
• $y^2=15 x^6+12 x^5+7 x^4+60 x^3+33 x^2+10 x+60$
• $y^2=x^6+x^3+23$
• $y^2=x^6+2 x^3+26$
• $y^2=61 x^6+53 x^5+51 x^4+60 x^3+6 x^2+66 x+32$
• $y^2=52 x^6+53 x^5+27 x^4+5 x^3+28 x^2+21 x+5$
• $y^2=54 x^6+61 x^5+66 x^4+52 x^3+49 x^2+66 x+38$
• $y^2=10 x^6+14 x^5+47 x^4+9 x^3+12 x^2+4 x+33$
• $y^2=14 x^6+66 x^5+63 x^4+27 x^3+46 x^2+44 x+2$
• $y^2=7 x^6+60 x^5+35 x^4+51 x^3+34 x^2+41 x+58$
• $y^2=53 x^6+41 x^5+66 x^4+66 x^3+62 x^2+45 x+25$
• $y^2=61 x^6+23 x^5+15 x^4+53 x^3+10 x^2+x+24$
• $y^2=33 x^6+53 x^5+10 x^4+56 x^3+26 x^2+50 x+34$
• $y^2=58 x^6+14 x^5+43 x^4+47 x^3+31 x^2+3 x+16$
• $y^2=62 x^6+41 x^5+64 x^4+66 x^3+36 x^2+x+20$
• $y^2=x^6+x^3+29$
• $y^2=45 x^6+51 x^5+21 x^4+22 x^3+61 x^2+60 x+7$
• $y^2=25 x^6+5 x^5+6 x^4+35 x^3+29 x^2+35 x+13$
• $y^2=62 x^6+46 x^5+49 x^4+x^3+17 x^2+4 x+37$
• $y^2=x^6+x^3+6$
• $y^2=44 x^6+2 x^5+34 x^4+2 x^3+18 x^2+48 x+5$
• $y^2=23 x^6+37 x^5+63 x^4+39 x^3+37 x^2+55 x+12$
• $y^2=x^6+2 x^3+36$

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.abcm^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.e_abz	$2$	(not in LMFDB)
2.67.i_fu	$3$	(not in LMFDB)
2.67.ai_fu	$6$	(not in LMFDB)