# Properties

 Label 5.3.ac_ab_e_i_abh Base field $\F_{3}$ Dimension $5$ $p$-rank $4$ Ordinary no Supersingular no Simple no Geometrically simple no Primitive yes Principally polarizable yes

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## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 + 3 x + 3 x^{2} )( 1 - 5 x + 11 x^{2} - 14 x^{3} + 17 x^{4} - 42 x^{5} + 99 x^{6} - 135 x^{7} + 81 x^{8} )$ $1 - 2 x - x^{2} + 4 x^{3} + 8 x^{4} - 33 x^{5} + 24 x^{6} + 36 x^{7} - 27 x^{8} - 162 x^{9} + 243 x^{10}$ Frobenius angles: $\pm0.0585233652746$, $\pm0.211357390051$, $\pm0.373008092126$, $\pm0.753920350296$, $\pm0.833333333333$ Angle rank: $4$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

## Newton polygon

 $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

## Point counts

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $91$ $36855$ $13632892$ $5042685375$ $638196694681$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $4$ $26$ $112$ $177$ $760$ $2179$ $6536$ $19781$ $58099$

## Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

## Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.d $\times$ 4.3.af_l_ao_r 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 $\times$ 4.729.ay_ceo_abigz_crind. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 4.729.ay_ceo_abigz_crind : 8.0.285524928949.1.
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 $\times$ 4.9.ad_p_abs_dz. 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 $\times$ 4.27.ac_ak_cd_wd. 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.

TwistExtension degreeCommon base change
5.3.ai_bd_ack_do_aff$2$(not in LMFDB)
5.3.c_ab_ae_i_bh$2$(not in LMFDB)
5.3.i_bd_ck_do_ff$2$(not in LMFDB)
5.3.ai_bd_ack_do_aff$3$(not in LMFDB)
5.3.af_o_abd_by_adg$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
5.3.ai_bd_ack_do_aff$2$(not in LMFDB)
5.3.c_ab_ae_i_bh$2$(not in LMFDB)
5.3.i_bd_ck_do_ff$2$(not in LMFDB)
5.3.ai_bd_ack_do_aff$3$(not in LMFDB)
5.3.af_o_abd_by_adg$3$(not in LMFDB)
5.3.f_o_bd_by_dg$6$(not in LMFDB)