Properties

Label 2.4.a_ah
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 no

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 7 x^{2} + 16 x^{4}$
Frobenius angles:  $\pm0.0804306232552$, $\pm0.919569376745$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{15})\)
Galois group:  $C_2^2$
Jacobians:  $0$
Isomorphism classes:  2

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $10$ $100$ $4090$ $57600$ $1050250$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $3$ $65$ $223$ $1025$ $4083$ $16385$ $65983$ $262145$ $1051923$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{15})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ah 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.ac_j$4$2.256.abi_bev
2.4.a_h$4$2.256.abi_bev
2.4.c_j$4$2.256.abi_bev

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

TwistExtension degreeCommon base change
2.4.ac_j$4$2.256.abi_bev
2.4.a_h$4$2.256.abi_bev
2.4.c_j$4$2.256.abi_bev
2.4.ab_ad$12$(not in LMFDB)
2.4.b_ad$12$(not in LMFDB)