Properties

Label 3.2.ae_j_ap
Base Field $\F_{2}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.0435981566527$, $\pm0.329312442367$, $\pm0.527830414776$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{7})\)
Galois group:  $C_6$
Jacobians:  0

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

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
$A(\F_{q^r})$ 1 71 421 2059 25621 209237 1560896 15222187 151446751 1147846421

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 7 8 7 24 52 90 231 575 1092

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{7})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{7}}$ is 1.128.an 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{7}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.e_j_p$2$3.4.c_ad_an
3.2.d_c_ab$7$(not in LMFDB)
3.2.d_j_n$7$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.e_j_p$2$3.4.c_ad_an
3.2.d_c_ab$7$(not in LMFDB)
3.2.d_j_n$7$(not in LMFDB)
3.2.ad_c_b$14$(not in LMFDB)
3.2.ad_j_an$14$(not in LMFDB)
3.2.ab_f_ad$14$(not in LMFDB)
3.2.b_f_d$14$(not in LMFDB)
3.2.e_j_p$14$(not in LMFDB)
3.2.a_a_af$21$(not in LMFDB)
3.2.ab_ab_d$28$(not in LMFDB)
3.2.b_ab_ad$28$(not in LMFDB)
3.2.ac_c_ad$42$(not in LMFDB)
3.2.a_a_f$42$(not in LMFDB)
3.2.c_c_d$42$(not in LMFDB)

Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.