Properties

Label 2.2.d_f
Base field F2\F_{2}
Dimension 22
pp-rank 22
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  F2\F_{2}
Dimension:  22
L-polynomial:  1+3x+5x2+6x3+4x41 + 3 x + 5 x^{2} + 6 x^{3} + 4 x^{4}
Frobenius angles:  ±0.543118021706\pm0.543118021706, ±0.876451355039\pm0.876451355039
Angle rank:  11 (numerical)
Number field:  Q(3,5)\Q(\sqrt{-3}, \sqrt{5})
Galois group:  C22C_2^2
Jacobians:  11
Isomorphism classes:  1

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.

pp-rank:  22
Slopes:  [0,0,1,1][0, 0, 1, 1]

Point counts

Point counts of the abelian variety

rr 11 22 33 44 55
A(Fqr)A(\F_{q^r}) 1919 1919 7676 171171 11591159

Point counts of the curve

rr 11 22 33 44 55 66 77 88 99 1010
C(Fqr)C(\F_{q^r}) 66 66 99 1010 3636 8787 9090 274274 513513 10861086

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over F26\F_{2^{6}}.

Endomorphism algebra over F2\F_{2}
The endomorphism algebra of this simple isogeny class is Q(3,5)\Q(\sqrt{-3}, \sqrt{5}).
Endomorphism algebra over F2\overline{\F}_{2}
The base change of AA to F26\F_{2^{6}} is 1.64.l 2 and its endomorphism algebra is M2(\mathrm{M}_{2}(Q(15)\Q(\sqrt{-15}) ))
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.2.ad_f222.4.b_ad
2.2.ad_f332.8.a_l
2.2.a_ab332.8.a_l

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.2.ad_f222.4.b_ad
2.2.ad_f332.8.a_l
2.2.a_ab332.8.a_l
2.2.a_b1212(not in LMFDB)