Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 9 x^{2} )( 1 - 8 x + 31 x^{2} - 72 x^{3} + 81 x^{4} )$ |
$1 - 12 x + 72 x^{2} - 268 x^{3} + 648 x^{4} - 972 x^{5} + 729 x^{6}$ | |
Frobenius angles: | $\pm0.0954872438962$, $\pm0.267720472801$, $\pm0.376614839446$ |
Angle rank: | $3$ (numerical) |
Isomorphism classes: | 20 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $198$ | $534996$ | $420216984$ | $286629456960$ | $205017018156918$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $82$ | $790$ | $6658$ | $58798$ | $530296$ | $4782958$ | $43057666$ | $387458422$ | $3486876082$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The isogeny class factors as 1.9.ae $\times$ 2.9.ai_bf and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
3.9.ae_i_au | $2$ | (not in LMFDB) |
3.9.e_i_u | $2$ | (not in LMFDB) |
3.9.m_cu_ki | $2$ | (not in LMFDB) |