Properties

Label 2.4.ae_i
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:  $( 1 - 2 x )^{2}( 1 + 4 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.5$
Angle rank:  $0$ (numerical)
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 5 225 3185 50625 985025 16769025 264273665 4228250625 68451564545 1099509530625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 17 49 193 961 4097 16129 64513 261121 1048577

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae $\times$ 1.4.a 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 $\F_{2^{8}}$ is 1.256.abg 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 $\F_{2^{8}}$.
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.e_i$2$2.16.a_abg
2.4.c_i$3$2.64.aq_ey
2.4.ai_y$4$2.256.acm_chc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.e_i$2$2.16.a_abg
2.4.c_i$3$2.64.aq_ey
2.4.ai_y$4$2.256.acm_chc
2.4.a_ai$4$2.256.acm_chc
2.4.a_i$4$2.256.acm_chc
2.4.i_y$4$2.256.acm_chc
2.4.ac_i$6$(not in LMFDB)
2.4.a_a$8$(not in LMFDB)
2.4.ag_q$12$(not in LMFDB)
2.4.ae_m$12$(not in LMFDB)
2.4.ac_a$12$(not in LMFDB)
2.4.a_ae$12$(not in LMFDB)
2.4.a_e$12$(not in LMFDB)
2.4.c_a$12$(not in LMFDB)
2.4.e_m$12$(not in LMFDB)
2.4.g_q$12$(not in LMFDB)
2.4.ac_e$20$(not in LMFDB)
2.4.c_e$20$(not in LMFDB)