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

• Label: 2.67.abc_mp
• Base field: $\F_{67}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$1 - 28 x + 327 x^{2} - 1876 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.0892044657248$, $\pm0.230349614374$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-69 +28 \sqrt{3}})\)
Galois group:	$D_{4}$
Jacobians:	$8$
Cyclic group of points:	yes

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2913$	$19578273$	$90424844964$	$406164449562729$	$1822909217259434793$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$4360$	$300652$	$20155924$	$1350178000$	$90458666350$	$6060711778000$	$406067667133924$	$27206534343683284$	$1822837805535119800$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=13 x^6+59 x^5+35 x^4+35 x^3+47 x^2+44 x+63$
• $y^2=60 x^6+41 x^5+65 x^4+7 x^3+30 x^2+60 x+66$
• $y^2=44 x^6+28 x^5+7 x^4+64 x^3+42 x^2+64 x+27$
• $y^2=7 x^6+38 x^5+35 x^4+25 x^3+64 x^2+23 x+43$
• $y^2=39 x^6+52 x^5+29 x^4+10 x^3+x^2+64 x+43$
• $y^2=46 x^6+10 x^5+64 x^4+7 x^3+66 x^2+39 x+27$
• $y^2=54 x^6+11 x^5+12 x^4+34 x^3+48 x^2+29 x+18$
• $y^2=20 x^6+8 x^5+56 x^4+64 x^3+21 x^2+53 x+11$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-69 +28 \sqrt{3}})\).

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
2.67.bc_mp	$2$	(not in LMFDB)