Properties

Label 2.4.ab_ad
Base field $\F_{2^{2}}$
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:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - x - 3 x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.0862360434115$, $\pm0.752902710078$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:  $C_2^2$
Jacobians:  $1$
Isomorphism classes:  3

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9$ $171$ $2916$ $69939$ $988749$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $10$ $43$ $274$ $964$ $4111$ $16636$ $65314$ $263707$ $1050250$

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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.4.b_ad$2$2.16.ah_bh
2.4.c_j$3$2.64.aw_jp
2.4.ac_j$6$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.4.b_ad$2$2.16.ah_bh
2.4.c_j$3$2.64.aw_jp
2.4.ac_j$6$(not in LMFDB)
2.4.a_h$6$(not in LMFDB)
2.4.a_ah$12$(not in LMFDB)