# Properties

 Label 4.3.ah_v_abl_cf Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 6 x^{2} - 7 x^{3} + 18 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.0452398905210$, $\pm0.166666666667$, $\pm0.239335307006$, $\pm0.691360448188$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 5 3465 321020 57882825 3993666025 274969038960 23543819129605 1809002263293225 146011266846851840 12156510723031191825

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 3 15 107 277 711 2251 6403 19140 59043

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 3.3.ae_g_ah 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 $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc $\times$ 3.729.acv_eli_afacr. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 3.729.acv_eli_afacr : 6.0.2461019.1.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 3.9.ae_q_acp. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 3.27.an_bw_acv. 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 4.3.b_ad_b_p $2$ (not in LMFDB) 4.3.h_v_bl_cf $2$ (not in LMFDB) 4.3.ae_j_at_bk $3$ (not in LMFDB) 4.3.ab_ad_ab_p $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.b_ad_b_p $2$ (not in LMFDB) 4.3.h_v_bl_cf $2$ (not in LMFDB) 4.3.ae_j_at_bk $3$ (not in LMFDB) 4.3.ab_ad_ab_p $3$ (not in LMFDB) 4.3.ae_j_at_bk $6$ (not in LMFDB) 4.3.e_j_t_bk $6$ (not in LMFDB)