# Properties

 Label 3.7.aj_bp_aew Base Field $\F_{7}$ Dimension $3$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{7}$ Dimension: $3$ L-polynomial: $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )( 1 + 7 x^{2} )$ Frobenius angles: $\pm0.106147807505$, $\pm0.227185525829$, $\pm0.5$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $2$ Slopes: $[0, 0, 1/2, 1/2, 1, 1]$

## 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 $A(\F_{q^r})$ 96 119808 40569984 13681115136 4822242966816 1645923601760256 559054007578523616 191429583261696000000 65712360862558465840512 22542892859348070835811328

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 51 344 2375 17069 118908 824291 5760239 40353608 282519771

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 1.7.ae $\times$ 1.7.a 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 $\F_{7^{6}}$ is 1.117649.la 2 $\times$ 1.117649.bak. The endomorphism algebra for each factor is: 1.117649.la 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.117649.bak : the quaternion algebra over $$\Q$$ ramified at $7$ and $\infty$.
All geometric endomorphisms are defined over $\F_{7^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{7^{2}}$  The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.ac $\times$ 1.49.o. The endomorphism algebra for each factor is: 1.49.al : $$\Q(\sqrt{-3})$$. 1.49.ac : $$\Q(\sqrt{-3})$$. 1.49.o : the quaternion algebra over $$\Q$$ ramified at $7$ and $\infty$.
• Endomorphism algebra over $\F_{7^{3}}$  The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.a $\times$ 1.343.u. 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.ab_b_ao $2$ (not in LMFDB) 3.7.b_b_o $2$ (not in LMFDB) 3.7.ag_ba_adg $3$ (not in LMFDB) 3.7.ad_r_abq $3$ (not in LMFDB) 3.7.a_ae_a $3$ (not in LMFDB) 3.7.a_f_a $3$ (not in LMFDB) 3.7.a_u_a $3$ (not in LMFDB) 3.7.d_r_bq $3$ (not in LMFDB) 3.7.g_ba_dg $3$ (not in LMFDB) 3.7.j_bp_ew $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.ab_b_ao $2$ (not in LMFDB) 3.7.b_b_o $2$ (not in LMFDB) 3.7.ag_ba_adg $3$ (not in LMFDB) 3.7.ad_r_abq $3$ (not in LMFDB) 3.7.a_ae_a $3$ (not in LMFDB) 3.7.a_f_a $3$ (not in LMFDB) 3.7.a_u_a $3$ (not in LMFDB) 3.7.d_r_bq $3$ (not in LMFDB) 3.7.g_ba_dg $3$ (not in LMFDB) 3.7.j_bp_ew $3$ (not in LMFDB) 3.7.ak_bu_afk $6$ (not in LMFDB) 3.7.ai_bl_aei $6$ (not in LMFDB) 3.7.af_z_acs $6$ (not in LMFDB) 3.7.ae_q_ace $6$ (not in LMFDB) 3.7.ac_w_abc $6$ (not in LMFDB) 3.7.c_w_bc $6$ (not in LMFDB) 3.7.d_r_bq $6$ (not in LMFDB) 3.7.e_q_ce $6$ (not in LMFDB) 3.7.f_z_cs $6$ (not in LMFDB) 3.7.i_bl_ei $6$ (not in LMFDB) 3.7.k_bu_fk $6$ (not in LMFDB) 3.7.a_ag_a $12$ (not in LMFDB) 3.7.a_j_a $12$ (not in LMFDB) 3.7.a_s_a $12$ (not in LMFDB)