Properties

Label 2.4.ac_j
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - x + 4 x^{2} )^{2}$
Frobenius angles:  $\pm0.419569376745$, $\pm0.419569376745$
Angle rank:  $1$ (numerical)
Jacobians:  1

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 16 576 5776 57600 929296 16842816 276756496 4324377600 68311140496 1096007985216

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 31 87 223 903 4111 16887 65983 260583 1045231

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.a_ab

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_h$2$2.16.o_dd
2.4.c_j$2$2.16.o_dd
2.4.b_ad$3$2.64.w_jp
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_h$2$2.16.o_dd
2.4.c_j$2$2.16.o_dd
2.4.b_ad$3$2.64.w_jp
2.4.a_ah$4$2.256.abi_bev
2.4.ab_ad$6$(not in LMFDB)