Properties

 Label 3.7.ak_bt_aff Base Field $\F_{7}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

Invariants

 Base field: $\F_{7}$ Dimension: $3$ L-polynomial: $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 13 x^{2} - 35 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.0616448849068$, $\pm0.106147807505$, $\pm0.511587336964$ Angle rank: $3$ (numerical)

This isogeny class is not 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 $A(\F_{q^r})$ 69 92391 34480404 12787745919 4727062720944 1637900701675056 558825243763515063 191566927803691316175 65743058244179560877892 22545439165532314764884736

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 40 289 2212 16733 118333 823954 5764372 40372453 282551675

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 2.7.af_n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7}$.

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 3.7.a_af_af $2$ (not in LMFDB) 3.7.a_af_f $2$ (not in LMFDB) 3.7.k_bt_ff $2$ (not in LMFDB) 3.7.ae_p_acf $3$ (not in LMFDB) 3.7.ab_a_as $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.a_af_af $2$ (not in LMFDB) 3.7.a_af_f $2$ (not in LMFDB) 3.7.k_bt_ff $2$ (not in LMFDB) 3.7.ae_p_acf $3$ (not in LMFDB) 3.7.ab_a_as $3$ (not in LMFDB) 3.7.aj_bo_aes $6$ (not in LMFDB) 3.7.ag_z_adf $6$ (not in LMFDB) 3.7.b_a_s $6$ (not in LMFDB) 3.7.e_p_cf $6$ (not in LMFDB) 3.7.g_z_df $6$ (not in LMFDB) 3.7.j_bo_es $6$ (not in LMFDB)