# Properties

 Label 3.2.ad_c_b 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 yes

## 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:

• $x^4+x^3y+x^2yz+xz^3+y^4+y^3z+z^4=0$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $29$ $301$ $8149$ $34861$

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

## 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)