Properties

Label 3.2.ad_c_b
Base Field $\F_{2}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.757883870938$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{7})\)
Galois group:  $C_6$
Jacobians:  1

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 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 29 301 8149 34861 375347 2863288 16436533 149808001 1062528419

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 0 3 28 35 87 168 252 570 1015

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.n 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.d_c_ab$2$3.4.af_s_abp
3.2.ad_j_an$7$(not in LMFDB)
3.2.e_j_p$7$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.d_c_ab$2$3.4.af_s_abp
3.2.ad_j_an$7$(not in LMFDB)
3.2.e_j_p$7$(not in LMFDB)
3.2.ae_j_ap$14$(not in LMFDB)
3.2.ab_f_ad$14$(not in LMFDB)
3.2.b_f_d$14$(not in LMFDB)
3.2.d_j_n$14$(not in LMFDB)
3.2.a_a_f$21$(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_af$42$(not in LMFDB)
3.2.c_c_d$42$(not in LMFDB)

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.