Invariants
Base field: | $\F_{3^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 16 x + 81 x^{2} )( 1 - 13 x + 81 x^{2} )$ |
$1 - 29 x + 370 x^{2} - 2349 x^{3} + 6561 x^{4}$ | |
Frobenius angles: | $\pm0.151478024726$, $\pm0.243120792737$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $16$ |
Isomorphism classes: | 80 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4554$ | $42397740$ | $282831106536$ | $1853767579023360$ | $12158284207270731354$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $53$ | $6461$ | $532196$ | $43064081$ | $3486961853$ | $282430693538$ | $22876796182013$ | $1853020171765601$ | $150094634976157796$ | $12157665457237837901$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^3+a^2+2a+2)x^6+(a+1)x^5+(a^3+a^2+a)x^4+(2a^2+2a)x^3+(2a^3+2)x^2+(a^2+a)x+2a^3+2a^2+2a+2$
- $y^2=(a^3+a^2+2a)x^6+(a^3+a^2+2a)x^5+(2a^3+2a+2)x^4+(2a^3+a^2+a)x^3+(a^2+1)x^2+(2a^3+2a^2+2a+2)x+2a$
- $y^2=(a^3+a+1)x^6+(a^3+2a)x^5+(2a^3+a^2+a+2)x^4+(a^2+2a)x^3+(2a^3+2a^2+2a+1)x^2+(2a^3+a^2+1)x+a^3+2$
- $y^2=(a^3+2a^2)x^6+(a^3+a^2+2a+2)x^5+(a^3+2a^2+a+2)x^4+(a^3+a^2+a+1)x^3+ax^2+(a^2+2a+2)x+a^3+2a^2+2a+2$
- $y^2=(a^2+2)x^6+2a^3x^5+(2a^2+2a+2)x^4+(2a^2+a+1)x^3+(2a^3+a^2)x^2+(2a^3+a+1)x+a^3+a^2+1$
- $y^2=(2a^3+a^2+2a+2)x^6+(a^3+2a^2+a)x^5+(a+1)x^4+(2a^3+2a^2+2a+2)x^3+(a^3+2a^2+2a+1)x^2+(a^3+2a^2+1)x+a^3+2a^2+a$
- $y^2=(a^3+a^2+2a+1)x^6+(a^3+a^2)x^5+(a^3+2a+1)x^4+(2a^2+a+2)x^3+(a^3+2a^2)x^2+(a^2+2)x+a^3+a^2+a$
- $y^2=(2a^3+a^2+2a+2)x^6+(a^3+2a^2+a)x^5+(2a^3+2a^2+a+1)x^4+(a^2+a+2)x^3+(2a^2+2a+2)x^2+(a^3+2a^2+2a+2)x+a^3+2a^2+a+1$
- $y^2=a^3x^6+(2a^3+2a^2+a)x^5+(2a^2+2a+1)x^4+(2a+2)x^3+(2a^3+a+1)x^2+(2a^2+a+2)x+a^2+a+1$
- $y^2=(a^3+2a^2+a+1)x^6+(2a^3+a+2)x^5+(a^3+a^2)x^4+(2a^3+a^2+2)x^3+(a^3+2a^2+2)x^2+(a^3+2a^2+2a)x+a^3+1$
- $y^2=(a^3+a^2+a+2)x^6+(a+2)x^5+(2a^2+2a+1)x^4+(a^3+2a^2+a+2)x^3+(2a^2+2a+1)x^2+2a^2x+a^3+2a^2+a$
- $y^2=(2a^2+a)x^6+(a^2+a+2)x^5+(2a+1)x^4+(a^3+2a^2+1)x^3+(a+2)x^2+2x+2a^2+1$
- $y^2=(a^3+2a^2+2a+2)x^6+(a^3+a^2+a+1)x^5+(2a^3+2a^2+2a)x^4+(a^2+a+2)x^3+(a^3+2a^2+a+1)x^2+(a^3+2a^2+2a+2)x$
- $y^2=(2a^2+a+2)x^6+(2a^3+a^2+a)x^5+(2a^2+2a+2)x^4+(a^3+a^2+a+1)x^3+(a+1)x^2+(2a+1)x+a^2+2a$
- $y^2=(a^3+2a+2)x^6+(2a^3+a^2+1)x^5+(2a^3+a+2)x^4+(2a^3+a^2+2a)x^3+2ax^2+a^3x+2a^3+a^2$
- $y^2=(2a^3+2a+2)x^6+2a^2x^5+(2a^3+a^2+a+2)x^4+(2a^3+a^2+1)x^3+(2a^3+2a+2)x^2+(a^2+1)x+2a^3+2a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3^{4}}$The isogeny class factors as 1.81.aq $\times$ 1.81.an 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 |
---|---|---|
2.81.ad_abu | $2$ | (not in LMFDB) |
2.81.d_abu | $2$ | (not in LMFDB) |
2.81.bd_og | $2$ | (not in LMFDB) |