# Properties

 Label 3.7.aj_bj_ads 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: $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 8 x^{2} - 28 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.0704914820143$, $\pm0.106147807505$, $\pm0.570491482014$ 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 $A(\F_{q^r})$ 78 91260 32760936 13026452400 4787838907098 1632410608570560 558141140442602886 191787062249502720000 65760214491382195402488 22541972682167303266921500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 39 272 2255 16949 117936 822947 5770991 40382984 282508239

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 2.7.ae_i 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.ack 2 $\times$ 1.2401.ax. The endomorphism algebra for each factor is: 1.2401.ack 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-10})$$$)$ 1.2401.ax : $$\Q(\sqrt{-3})$$.
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.al $\times$ 2.49.a_ack. 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_af_q $2$ (not in LMFDB) 3.7.b_af_aq $2$ (not in LMFDB) 3.7.j_bj_ds $2$ (not in LMFDB) 3.7.ad_l_abw $3$ (not in LMFDB) 3.7.a_ab_ay $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.ab_af_q $2$ (not in LMFDB) 3.7.b_af_aq $2$ (not in LMFDB) 3.7.j_bj_ds $2$ (not in LMFDB) 3.7.ad_l_abw $3$ (not in LMFDB) 3.7.a_ab_ay $3$ (not in LMFDB) 3.7.ai_bf_adk $6$ (not in LMFDB) 3.7.af_t_acm $6$ (not in LMFDB) 3.7.a_ab_y $6$ (not in LMFDB) 3.7.d_l_bw $6$ (not in LMFDB) 3.7.f_t_cm $6$ (not in LMFDB) 3.7.i_bf_dk $6$ (not in LMFDB) 3.7.af_b_be $8$ (not in LMFDB) 3.7.af_n_abe $8$ (not in LMFDB) 3.7.f_b_abe $8$ (not in LMFDB) 3.7.f_n_be $8$ (not in LMFDB) 3.7.ae_b_y $24$ (not in LMFDB) 3.7.ae_n_ay $24$ (not in LMFDB) 3.7.ab_b_g $24$ (not in LMFDB) 3.7.ab_n_ag $24$ (not in LMFDB) 3.7.b_b_ag $24$ (not in LMFDB) 3.7.b_n_g $24$ (not in LMFDB) 3.7.e_b_ay $24$ (not in LMFDB) 3.7.e_n_y $24$ (not in LMFDB)