# Properties

 Label 3.7.ak_bx_aga 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: $( 1 - 4 x + 7 x^{2} )( 1 - 6 x + 18 x^{2} - 42 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.0461154155528$, $\pm0.227185525829$, $\pm0.453884584447$ Angle rank: $2$ (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 $A(\F_{q^r})$ 80 111360 40629680 13434470400 4710799682000 1632390679745280 558596627678473520 191319632635310899200 65666478359851384193360 22537578552502661651424000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 48 346 2332 16678 117936 823618 5756924 40325422 282453168

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.ae $\times$ 2.7.ag_s and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{7}$
 The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ade 2 $\times$ 1.2401.dq. The endomorphism algebra for each factor is: 1.2401.ade 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$ 1.2401.dq : $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{7^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{7^{2}}$  The base change of $A$ to $\F_{7^{2}}$ is 1.49.ac $\times$ 2.49.a_ade. The endomorphism algebra for each factor is:

## 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.ac_b_am $2$ (not in LMFDB) 3.7.c_b_m $2$ (not in LMFDB) 3.7.k_bx_ga $2$ (not in LMFDB) 3.7.ah_bf_ady $3$ (not in LMFDB) 3.7.ab_af_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.ac_b_am $2$ (not in LMFDB) 3.7.c_b_m $2$ (not in LMFDB) 3.7.k_bx_ga $2$ (not in LMFDB) 3.7.ah_bf_ady $3$ (not in LMFDB) 3.7.ab_af_g $3$ (not in LMFDB) 3.7.c_b_m $4$ (not in LMFDB) 3.7.al_cd_ags $6$ (not in LMFDB) 3.7.af_t_aco $6$ (not in LMFDB) 3.7.b_af_ag $6$ (not in LMFDB) 3.7.f_t_co $6$ (not in LMFDB) 3.7.h_bf_dy $6$ (not in LMFDB) 3.7.l_cd_gs $6$ (not in LMFDB) 3.7.ae_d_q $8$ (not in LMFDB) 3.7.ae_l_aq $8$ (not in LMFDB) 3.7.e_d_aq $8$ (not in LMFDB) 3.7.e_l_q $8$ (not in LMFDB) 3.7.af_d_u $24$ (not in LMFDB) 3.7.af_l_au $24$ (not in LMFDB) 3.7.ab_d_e $24$ (not in LMFDB) 3.7.ab_l_ae $24$ (not in LMFDB) 3.7.b_d_ae $24$ (not in LMFDB) 3.7.b_l_e $24$ (not in LMFDB) 3.7.f_d_au $24$ (not in LMFDB) 3.7.f_l_u $24$ (not in LMFDB)