Properties

Label 2.4.a_ah
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
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

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 10 100 4090 57600 1050250 16728100 268465690 4324377600 68719562410 1103025062500

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

Decomposition and endomorphism algebra

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.

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)