Properties

Label 2.2.b_a
Base field F2\F_{2}
Dimension 22
pp-rank 11
Ordinary no
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  F2\F_{2}
Dimension:  22
L-polynomial:  1+x+2x3+4x41 + x + 2 x^{3} + 4 x^{4}
Frobenius angles:  ±0.313829601922\pm0.313829601922, ±0.860613258134\pm0.860613258134
Angle rank:  22 (numerical)
Number field:  4.0.2312.1
Galois group:  D4D_{4}
Jacobians:  11
Isomorphism classes:  1

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

pp-rank:  11
Slopes:  [0,1/2,1/2,1][0, 1/2, 1/2, 1]

Point counts

Point counts of the abelian variety

rr 11 22 33 44 55
A(Fqr)A(\F_{q^r}) 88 1616 152152 416416 808808

Point counts of the curve

rr 11 22 33 44 55 66 77 88 99 1010
C(Fqr)C(\F_{q^r}) 44 44 1616 2424 2424 6464 8888 288288 520520 11041104

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 F2\F_{2}.

Endomorphism algebra over F2\F_{2}
The endomorphism algebra of this simple isogeny class is 4.0.2312.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.2.ab_a222.4.ab_e