Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 10 x + 27 x^{2} )( 1 - 8 x + 27 x^{2} )$ |
$1 - 18 x + 134 x^{2} - 486 x^{3} + 729 x^{4}$ | |
Frobenius angles: | $\pm0.0877398280459$, $\pm0.220355751984$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 16 |
This isogeny class is not simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $360$ | $492480$ | $386371080$ | $282801715200$ | $205973541793800$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $674$ | $19630$ | $532142$ | $14354650$ | $387444626$ | $10460397790$ | $282429385118$ | $7625596330090$ | $205891135662914$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+1)x^6+(a^2+2a)x^5+(2a^2+2a+2)x^4+ax^3+(2a^2+2a+2)x^2+(a^2+2a)x+a+1$
- $y^2=2x^6+(a+2)x^5+2a^2x^4+(2a+1)x^3+2a^2x^2+(a+2)x+2$
- $y^2=(a+1)x^6+(a^2+2a)x^5+x^4+(a^2+a)x^3+(a^2+a)x^2+(2a^2+2a+1)x+2$
- $y^2=(2a^2+1)x^6+(a^2+a)x^5+(2a^2+2a+1)x^4+a^2x^3+a^2x^2+(a^2+2a+1)x+a^2+a+2$
- $y^2=(a^2+2a+2)x^6+(2a^2+a)x^5+ax^4+ax^3+ax^2+(2a^2+a)x+a^2+2a+2$
- $y^2=(a^2+1)x^6+a^2x^5+(a^2+2a+1)x^4+(a^2+2a)x^3+(a^2+2a)x^2+x+2a^2+a+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{3}}$.
Endomorphism algebra over $\F_{3^{3}}$The isogeny class factors as 1.27.ak $\times$ 1.27.ai 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.
Subfield | Primitive Model |
$\F_{3}$ | 2.3.d_i |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.27.ac_aba | $2$ | 2.729.ace_cvu |
2.27.c_aba | $2$ | 2.729.ace_cvu |
2.27.s_fe | $2$ | 2.729.ace_cvu |