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

• Label: 2.67.abc_mr
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$( 1 - 15 x + 67 x^{2} )( 1 - 13 x + 67 x^{2} )$
	$1 - 28 x + 329 x^{2} - 1876 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.131184157393$, $\pm0.207941879321$
Angle rank:	$2$ (numerical)
Jacobians:	$9$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2915$	$19597545$	$90475676720$	$406238000243625$	$1822982937421226075$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$4364$	$300820$	$20159572$	$1350232600$	$90459282566$	$6060717052360$	$406067697566308$	$27206534371102060$	$1822837803439942364$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):

• $y^2=41 x^6+65 x^5+25 x^4+6 x^3+26 x^2+36 x+7$
• $y^2=2 x^6+21 x^5+30 x^4+47 x^3+58 x^2+24 x+51$
• $y^2=4 x^6+4 x^5+19 x^4+62 x^3+55 x^2+17 x+21$
• $y^2=58 x^6+25 x^5+6 x^4+63 x^3+40 x^2+54 x+58$
• $y^2=45 x^6+17 x^5+3 x^4+61 x^3+63 x^2+60 x+5$
• $y^2=31 x^6+32 x^5+4 x^4+6 x^2+5 x+46$
• $y^2=42 x^6+23 x^5+29 x^4+48 x^3+64 x^2+40 x+5$
• $y^2=41 x^6+x^4+22 x^3+37 x^2+41$
• $y^2=18 x^6+48 x^5+55 x^4+55 x^3+17 x^2+7 x+57$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.ap $\times$ 1.67.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.67.ap : \(\Q(\sqrt{-43}) \).
• 1.67.an : \(\Q(\sqrt{-11}) \).

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.ac_acj	$2$	(not in LMFDB)
2.67.c_acj	$2$	(not in LMFDB)
2.67.bc_mr	$2$	(not in LMFDB)