Properties

Label 2.4.a_e
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 2 x + 4 x^{2} )( 1 + 2 x + 4 x^{2} )$
Frobenius angles:  $\pm0.333333333333$, $\pm0.666666666667$
Angle rank:  $0$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21 441 3969 74529 1049601 15752961 268451841 4328718849 68718952449 1101662259201

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 25 65 289 1025 3841 16385 66049 262145 1050625

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ac $\times$ 1.4.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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
2.4.ae_m$2$2.16.i_bw
2.4.e_m$2$2.16.i_bw
2.4.ag_q$3$2.64.a_aey
2.4.a_ai$3$2.64.a_aey
2.4.g_q$3$2.64.a_aey
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ae_m$2$2.16.i_bw
2.4.e_m$2$2.16.i_bw
2.4.ag_q$3$2.64.a_aey
2.4.a_ai$3$2.64.a_aey
2.4.g_q$3$2.64.a_aey
2.4.a_ae$4$2.256.bg_bdo
2.4.ai_y$6$(not in LMFDB)
2.4.ac_a$6$(not in LMFDB)
2.4.c_a$6$(not in LMFDB)
2.4.i_y$6$(not in LMFDB)
2.4.ae_i$12$(not in LMFDB)
2.4.ac_i$12$(not in LMFDB)
2.4.a_ae$12$(not in LMFDB)
2.4.a_i$12$(not in LMFDB)
2.4.c_i$12$(not in LMFDB)
2.4.e_i$12$(not in LMFDB)
2.4.a_a$24$(not in LMFDB)
2.4.ac_e$30$(not in LMFDB)
2.4.c_e$30$(not in LMFDB)