Invariants
Base field: | $\F_{3^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 29 x + 243 x^{2} )( 1 - 28 x + 243 x^{2} )$ |
$1 - 57 x + 1298 x^{2} - 13851 x^{3} + 59049 x^{4}$ | |
Frobenius angles: | $\pm0.119654564389$, $\pm0.144947286894$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $46440$ | $3448448640$ | $205822463222880$ | $12157739967920601600$ | $717899473696939086742200$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $187$ | $58397$ | $14344120$ | $3486805769$ | $847290363277$ | $205891176569174$ | $50031545936253271$ | $12157665472083090641$ | $2954312706719024345800$ | $717897987693502120524557$ |
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 $\F_{3^{5}}$.
Endomorphism algebra over $\F_{3^{5}}$The isogeny class factors as 1.243.abd $\times$ 1.243.abc 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 |
---|---|---|
2.243.ab_amo | $2$ | (not in LMFDB) |
2.243.b_amo | $2$ | (not in LMFDB) |
2.243.cf_bxy | $2$ | (not in LMFDB) |