# 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

## 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 6 7 8 9 10 $A(\F_{q^r})$ 1 71 421 2059 25621 209237 1560896 15222187 151446751 1147846421

 $r$ 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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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)