Properties

Label 2.2.ad_f
Base field $\F_{2}$
Dimension $2$
$p$-rank $2$
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:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4}$
Frobenius angles:  $\pm0.123548644961$, $\pm0.456881978294$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:  $C_2^2$
Jacobians:  $1$
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.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $19$ $76$ $171$ $961$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $6$ $9$ $10$ $30$ $87$ $168$ $274$ $513$ $1086$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.l 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
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.d_f$2$2.4.b_ad
2.2.a_ab$3$2.8.a_l
2.2.d_f$3$2.8.a_l

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

TwistExtension degreeCommon base change
2.2.d_f$2$2.4.b_ad
2.2.a_ab$3$2.8.a_l
2.2.d_f$3$2.8.a_l
2.2.a_ab$6$2.64.w_jp
2.2.a_b$12$(not in LMFDB)

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1.