Properties

Label 4.2.ae_i_am_r
Base Field $\F_{2}$
Dimension $4$
Ordinary Yes
$p$-rank $4$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2}$
Dimension:  $4$
L-polynomial:  $1 - 4 x + 8 x^{2} - 12 x^{3} + 17 x^{4} - 24 x^{5} + 32 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.0755571399449$, $\pm0.203216343788$, $\pm0.424442860055$, $\pm0.703216343788$
Angle rank:  $2$ (numerical)
Number field:  8.0.18939904.2
Galois group:  $D_4\times C_2$

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $4$
Slopes:  $[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2 292 2402 85264 930082 17183908 403078034 4278206464 59240758994 1097334925732

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 5 5 21 29 65 181 253 437 1025

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is 8.0.18939904.2.
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{4}}$ is 2.16.c_b 2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1088.2$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.
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
4.2.e_i_m_r$2$4.4.a_c_a_b
4.2.ae_k_au_bh$8$(not in LMFDB)
4.2.a_ac_a_b$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.2.e_i_m_r$2$4.4.a_c_a_b
4.2.ae_k_au_bh$8$(not in LMFDB)
4.2.a_ac_a_b$8$(not in LMFDB)
4.2.a_c_a_b$8$(not in LMFDB)
4.2.e_k_u_bh$8$(not in LMFDB)
4.2.ac_b_c_ad$24$(not in LMFDB)
4.2.c_b_ac_ad$24$(not in LMFDB)