Abelian Variety Isogeny Class 2.64.a_abp over $\F_{2^{6}}$

• Label: 2.64.a_abp
• Base Field: $\F_{2^{6}}$
• Dimension: $2$
• Ordinary: No
• $p$-rank: $2$
• Principally polarizable: Yes
• Contains a Jacobian: Yes

Invariants

Base field:	$\F_{2^{6}}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x + 64 x^{2} )( 1 + 13 x + 64 x^{2} )$
Frobenius angles:	$\pm0.198106042756$, $\pm0.801893957244$
Angle rank:	$1$ (numerical)
Jacobians:	426

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 426 curves, and hence is principally polarizable:

• $y^2+(x^2+x)y=ax^5+(a^3+a^2+a)x^3+(a^2+1)x^2+(a^3+1)x$
• $y^2+(x^2+x)y=ax^5+(a^5+a^3+a^2+a)x^4+(a^3+a^2+a)x^3+(a^5+a^3+a+1)x^2+(a^3+1)x$
• $y^2+(x^2+x)y=(a^5+a^4+a^2+a+1)x^5+(a^3+a^2+a+1)x^3+(a^5+a^3+a)x^2+(a^4+a)x$
• $y^2+(x^2+x)y=(a^5+a^4+a^2+a+1)x^5+(a^4+a^3+1)x^4+(a^3+a^2+a+1)x^3+(a^5+a^4+a+1)x^2+(a^4+a)x$
• $y^2+(x^2+x)y=(a^4+a+1)x^5+(a^3+a^2+a)x^3+(a^5+a^4+a+1)x^2+(a^5+a^3+a^2+a)x$
• $y^2+(x^2+x)y=(a^4+a+1)x^5+(a^5+a^3+a^2)x^4+(a^3+a^2+a)x^3+(a^4+a^3+a^2+a+1)x^2+(a^5+a^3+a^2+a)x$
• $y^2+(x^2+x)y=a^2x^5+(a^3+a^2+a+1)x^3+(a^4+1)x^2+(a^4+a^3+a)x$
• $y^2+(x^2+x)y=a^2x^5+(a^5+a^3+1)x^4+(a^3+a^2+a+1)x^3+(a^5+a^4+a^3)x^2+(a^4+a^3+a)x$
• $y^2+(x^2+x)y=a^4x^5+(a^3+a^2+a)x^3+(a^5+a^3)x^2+(a^5+a^4+a^2+a)x$
• $y^2+(x^2+x)y=a^4x^5+(a^5+a^4+a^3+a+1)x^4+(a^3+a^2+a)x^3+(a^4+a+1)x^2+(a^5+a^4+a^2+a)x$
• $y^2+(x^2+x)y=(a^5+a^4+a)x^5+(a^3+a^2+a+1)x^3+(a+1)x^2+(a^5+a^4+a^3+a^2+a)x$
• $y^2+(x^2+x)y=(a^5+a^4+a)x^5+(a^4+a^3+a^2+1)x^4+(a^3+a^2+a+1)x^3+(a^4+a^3+a^2+a)x^2+(a^5+a^4+a^3+a^2+a)x$
• $y^2+(x^2+x+a^5+a^3+a)y=(a+1)x^5+(a+1)x^3+a^4x^2+(a^5+a^4+a^3+1)x+a^5+a+1$
• $y^2+(x^2+x+a^5+a^3+a)y=(a+1)x^5+(a^5+a^3+a)x^4+(a+1)x^3+(a^5+a^4+a^3+a)x^2+(a^5+a^4+a^3+1)x+a^4+a+1$
• $y^2+(x^2+x+a^5+a^3+a^2+a)y=(a^4+a^2+1)x^5+(a^5+a^4)x^3+(a^3+a)x+a^3+a+1$
• $y^2+(x^2+x+a^5+a^3+a^2+a)y=(a^4+a^2+1)x^5+(a^5+a^4+a^3+a)x^4+(a^5+a^4)x^3+(a^5+a^4+a^3+a)x^2+(a^3+a)x+a^4+a^2+a+1$
• $y^2+(x^2+x)y=(a^5+a^2+1)x^5+(a^5+a^4+a^3+a+1)x^3+(a^3+a+1)x^2+(a^4+a^2+1)x$
• $y^2+(x^2+x)y=(a^5+a^2+1)x^5+(a^3+a)x^4+(a^5+a^4+a^3+a+1)x^3+x^2+(a^4+a^2+1)x$
• $y^2+(x^2+x+a^5+a^4+a^3+a+1)y=(a^2+1)x^5+(a^2+1)x^3+(a^5+a^4+a^2+a+1)x^2+(a^4+a^2)x+a^4+a^3+a+1$
• $y^2+(x^2+x+a^5+a^4+a^3+a+1)y=(a^2+1)x^5+(a^5+a^4+a^3+a+1)x^4+(a^2+1)x^3+(a^3+a^2)x^2+(a^4+a^2)x+a^4+a^3+a^2+1$
• and 406 more

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	4056	16451136	68719911624	281693525577984	1152921502463163576	4722426253610370317376	19342813113840728124788136	79228157538525417810440193024	324518553658426719376279169811864	1329227990841918495641742802133107776

Point counts of the curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	65	4015	262145	16790239	1073741825	68720346511	4398046511105	281474959033279	18014398509481985	1152921500319480175

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$

The isogeny class factors as 1.64.an $\times$ 1.64.n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.64.an : \(\Q(\sqrt{-87}) \).
• 1.64.n : \(\Q(\sqrt{-87}) \).

Endomorphism algebra over $\overline{\F}_{2^{6}}$

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.abp^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-87}) \)$)$

All geometric endomorphisms are defined over $\F_{2^{12}}$.

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
2.64.aba_ll	$2$	(not in LMFDB)
2.64.ba_ll	$2$	(not in LMFDB)
2.64.a_bp	$4$	(not in LMFDB)