Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 13 x + 49 x^{2} )( 1 - 8 x + 49 x^{2} )$ |
$1 - 21 x + 202 x^{2} - 1029 x^{3} + 2401 x^{4}$ | |
Frobenius angles: | $\pm0.121037718324$, $\pm0.306389419005$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $16$ |
Isomorphism classes: | 72 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1554$ | $5678316$ | $13885804296$ | $33252581908224$ | $79794396406737474$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $29$ | $2365$ | $118028$ | $5768209$ | $282482789$ | $13841235106$ | $678223054181$ | $33232940278369$ | $1628413733857772$ | $79792267297206925$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(6a+1)x^6+(4a+6)x^5+5x^4+(5a+2)x^3+(a+3)x^2+ax+5a+3$
- $y^2=(6a+5)x^6+(6a+3)x^5+(6a+6)x^4+(5a+4)x^3+(5a+6)x^2+(3a+3)x+5a$
- $y^2=ax^6+(a+4)x^5+(6a+2)x^4+ax^3+6ax^2+(3a+5)x+5a+1$
- $y^2=(2a+3)x^6+(a+5)x^5+(3a+2)x^4+(a+2)x^3+(3a+5)x^2+(a+6)x+5a+5$
- $y^2=(4a+6)x^6+(a+2)x^5+5x^4+(3a+5)x^3+(6a+5)x^2+(2a+5)x+3$
- $y^2=(a+3)x^6+5ax^5+3x^4+2ax^3+(5a+4)x^2+(a+1)x+3a+3$
- $y^2=(2a+6)x^6+(a+1)x^5+3ax^4+(3a+5)x^3+(a+1)x^2+5x+4a+4$
- $y^2=6ax^6+(a+4)x^5+(3a+3)x^4+(6a+1)x^3+(4a+3)x^2+(a+6)x+a+4$
- $y^2=(a+6)x^6+(a+1)x^5+5ax^4+(a+4)x^3+5x^2+(4a+2)x+3a+4$
- $y^2=(5a+2)x^6+(6a+4)x^5+5x^4+(a+3)x^3+(6a+3)x^2+(a+1)x+3a+4$
- $y^2=(2a+6)x^6+(5a+2)x^5+(5a+2)x^4+(4a+1)x^3+(2a+6)x^2+(a+6)x+6a+1$
- $y^2=(a+5)x^6+(3a+2)x^5+(3a+2)x^4+(2a+1)x^3+(2a+4)x^2+ax+3a$
- $y^2=(5a+4)x^6+2x^5+(3a+3)x^4+(3a+4)x^3+(6a+6)x^2+(6a+3)x+a+2$
- $y^2=(3a+1)x^6+(3a+3)x^5+2ax^4+(5a+3)x^3+(4a+5)x^2+(4a+3)x+6a$
- $y^2=(6a+6)x^6+(2a+3)x^5+(2a+1)x^4+4x^3+3x^2+(6a+1)x+2a+3$
- $y^2=(2a+2)x^6+(2a+1)x^5+(4a+6)x^4+(4a+6)x^3+(4a+5)x^2+(6a+6)x+3a+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$The isogeny class factors as 1.49.an $\times$ 1.49.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 is a primitive isogeny class.