Properties

Label 2.4.ae_m
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:  $( 1 - 2 x + 4 x^{2} )^{2}$
Frobenius angles:  $\pm0.333333333333$, $\pm0.333333333333$
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
$A(\F_{q^r})$ 9 441 6561 74529 986049 15752961 264290049 4328718849 69257922561 1101662259201

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 25 97 289 961 3841 16129 66049 264193 1050625

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.q 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 $\F_{2^{6}}$.

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_ac

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_e$2$2.16.i_bw
2.4.e_m$2$2.16.i_bw
2.4.c_a$3$2.64.bg_ou
2.4.i_y$3$2.64.bg_ou
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_e$2$2.16.i_bw
2.4.e_m$2$2.16.i_bw
2.4.c_a$3$2.64.bg_ou
2.4.i_y$3$2.64.bg_ou
2.4.a_ae$4$2.256.bg_bdo
2.4.ai_y$6$(not in LMFDB)
2.4.ag_q$6$(not in LMFDB)
2.4.ac_a$6$(not in LMFDB)
2.4.a_ai$6$(not in LMFDB)
2.4.g_q$6$(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.ac_e$15$(not in LMFDB)
2.4.a_a$24$(not in LMFDB)
2.4.c_e$30$(not in LMFDB)