Properties

Label 2.4.a_ae
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:  $1 - 4 x^{2} + 16 x^{4}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.833333333333$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\zeta_{12})\)
Galois group:  $C_2^2$
Jacobians:  1

This isogeny class is simple but not geometrically 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
$A(\F_{q^r})$ 13 169 4225 74529 1047553 17850625 268419073 4328718849 68720001025 1097367287809

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 9 65 289 1025 4353 16385 66049 262145 1046529

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 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 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.ac_c
$\F_{2}$2.2.c_c

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_i$3$2.64.a_ey
2.4.ae_m$4$2.256.bg_bdo
2.4.a_e$4$2.256.bg_bdo
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_i$3$2.64.a_ey
2.4.ae_m$4$2.256.bg_bdo
2.4.a_e$4$2.256.bg_bdo
2.4.e_m$4$2.256.bg_bdo
2.4.a_i$6$(not in LMFDB)
2.4.ai_y$12$(not in LMFDB)
2.4.ag_q$12$(not in LMFDB)
2.4.ae_i$12$(not in LMFDB)
2.4.ae_m$12$(not in LMFDB)
2.4.ac_a$12$(not in LMFDB)
2.4.ac_i$12$(not in LMFDB)
2.4.a_ai$12$(not in LMFDB)
2.4.a_e$12$(not in LMFDB)
2.4.c_a$12$(not in LMFDB)
2.4.c_i$12$(not in LMFDB)
2.4.e_i$12$(not in LMFDB)
2.4.e_m$12$(not in LMFDB)
2.4.g_q$12$(not in LMFDB)
2.4.i_y$12$(not in LMFDB)
2.4.a_a$24$(not in LMFDB)
2.4.ac_e$60$(not in LMFDB)
2.4.c_e$60$(not in LMFDB)