# Properties

 Label 4.3.ai_bh_adn_hb 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: $( 1 - 3 x + 3 x^{2} )( 1 - 5 x + 15 x^{2} - 31 x^{3} + 45 x^{4} - 45 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.113296540390$, $\pm0.166666666667$, $\pm0.351823865540$, $\pm0.481790494592$ 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 $A(\F_{q^r})$ 7 8281 732844 40585181 3533486887 315571420528 24860761811648 1917568997154629 153059630035416076 12278172330073214201

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 12 35 76 246 813 2369 6788 20069 59632

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 3.3.af_p_abf 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.bd_apb_abvut. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 3.729.bd_apb_abvut : 6.0.400967.1.
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.f_f_ah. 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.h_bn_lb. 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.ac_d_ab_ad $2$ (not in LMFDB) 4.3.c_d_b_ad $2$ (not in LMFDB) 4.3.i_bh_dn_hb $2$ (not in LMFDB) 4.3.af_s_abu_dm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ac_d_ab_ad $2$ (not in LMFDB) 4.3.c_d_b_ad $2$ (not in LMFDB) 4.3.i_bh_dn_hb $2$ (not in LMFDB) 4.3.af_s_abu_dm $3$ (not in LMFDB) 4.3.f_s_bu_dm $6$ (not in LMFDB)