Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 10 x + 27 x^{2} )( 1 - 7 x + 27 x^{2} )$ |
$1 - 17 x + 124 x^{2} - 459 x^{3} + 729 x^{4}$ | |
Frobenius angles: | $\pm0.0877398280459$, $\pm0.264757707515$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $3$ |
Isomorphism classes: | 15 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $378$ | $502740$ | $388086552$ | $282841524000$ | $205927500801078$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $689$ | $19718$ | $532217$ | $14351441$ | $387412946$ | $10460241803$ | $282429175793$ | $7625599857746$ | $205891171523489$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^2+a)x^6+(2a^2+2)x^5+(2a^2+a)x^4+2a^2x^3+a^2x^2+(2a^2+1)x+a^2+a+2$
- $y^2=(2a^2+2a)x^6+(a^2+a+1)x^5+(2a+2)x^4+(a^2+2a+2)x^2+(2a^2+1)x+a^2+2$
- $y^2=(2a^2+2a)x^6+x^5+(2a^2+2a+2)x^4+(2a^2+2a+2)x^3+(a^2+a+1)x^2+(a^2+1)x+2a^2+1$
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.ah 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.ad_aq | $2$ | 2.729.abp_bvg |
2.27.d_aq | $2$ | 2.729.abp_bvg |
2.27.r_eu | $2$ | 2.729.abp_bvg |