Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 10 x + 27 x^{2} )( 1 - 5 x + 27 x^{2} )$ |
$1 - 15 x + 104 x^{2} - 405 x^{3} + 729 x^{4}$ | |
Frobenius angles: | $\pm0.0877398280459$, $\pm0.340228233311$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $9$ |
Isomorphism classes: | 39 |
This isogeny class is not simple, 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $414$ | $519156$ | $389178216$ | $282408404256$ | $205806150001314$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $713$ | $19774$ | $531401$ | $14342983$ | $387384722$ | $10460350669$ | $282430848593$ | $7625607263578$ | $205891167362153$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^2+2a+2)x^6+(2a^2+1)x^4+(2a^2+a+1)x^3+(2a^2+2)x^2+(a^2+a)x+2a^2+a+2$
- $y^2=(a^2+2a+2)x^6+(2a^2+1)x^5+(a^2+a+2)x^4+(a^2+2a+1)x^3+(2a^2+1)x^2+(a^2+2a)x+2a^2+2a$
- $y^2=ax^6+(2a^2+2a+1)x^5+(2a^2+2a+1)x^4+(2a^2+1)x^3+(a^2+1)x^2+(2a^2+a+1)x+a+1$
- $y^2=(a^2+1)x^6+2x^5+(a^2+a)x^4+(a^2+1)x^3+(2a^2+2a+2)x^2+2x+a$
- $y^2=(2a^2+1)x^6+(2a^2+2a+1)x^5+(a^2+2a)x^4+(2a^2+2a+1)x^3+(2a+1)x^2+a^2x+2$
- $y^2=(2a^2+a)x^6+x^5+a^2x^4+(2a^2+a+1)x^3+x^2+(2a^2+2a+1)x+a^2+2a+2$
- $y^2=(a^2+1)x^6+(2a^2+a)x^5+(a^2+2a)x^4+(2a+2)x^3+(a^2+2a+1)x^2+x+a$
- $y^2=(2a^2+2a+2)x^6+(a^2+1)x^5+(2a^2+1)x^4+(2a^2+a+1)x^3+(2a^2+a)x^2+(2a^2+2)x+2a^2+2a+1$
- $y^2=(a+1)x^6+(2a^2+2a+1)x^5+2a^2x^4+(2a^2+1)x^3+ax^2+(a^2+a+2)x+2a^2+a$
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.af 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 |
---|---|---|
2.27.af_e | $2$ | 2.729.ar_eu |
2.27.f_e | $2$ | 2.729.ar_eu |
2.27.p_ea | $2$ | 2.729.ar_eu |