# Properties

 Label 3.2.ae_j_ap Base field $\F_{2}$ Dimension $3$ $p$-rank $3$ Ordinary yes Supersingular no Simple yes Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian no

## 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.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $71$ $421$ $2059$ $25621$

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

## 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)