Properties

Label 2.2.ac_c
Base Field $\F_{2}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4}$
Frobenius angles:  $\pm0.0833333333333$, $\pm0.583333333333$
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
$A(\F_{q^r})$ 1 13 25 169 1321 4225 14449 74529 297025 1047553

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 5 1 9 41 65 113 289 577 1025

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).
Endomorphism algebra over $\overline{\F}_{2}$
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 is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.c_c$2$2.4.a_ae
2.2.e_i$3$2.8.ai_bg
2.2.ae_i$6$2.64.a_ey
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.c_c$2$2.4.a_ae
2.2.e_i$3$2.8.ai_bg
2.2.ae_i$6$2.64.a_ey
2.2.a_a$6$2.64.a_ey
2.2.a_ac$8$2.256.bg_bdo
2.2.a_c$8$2.256.bg_bdo
2.2.ac_e$24$(not in LMFDB)
2.2.a_ae$24$(not in LMFDB)
2.2.a_e$24$(not in LMFDB)
2.2.c_e$24$(not in LMFDB)

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1.