Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 21 x + 197 x^{2} - 1029 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.0459942676841$, $\pm0.329489468851$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.14725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1549$ | $5652301$ | $13848507661$ | $33224728332789$ | $79780901567145424$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $29$ | $2355$ | $117713$ | $5763379$ | $282435014$ | $13840900251$ | $678221257841$ | $33232930697539$ | $1628413656587957$ | $79792266566061150$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5ax^6+(a+5)x^5+(a+2)x^4+(3a+3)x^3+(5a+2)x^2+(6a+1)x+5a+6$
- $y^2=(a+1)x^6+(6a+3)x^4+(6a+5)x^3+(6a+5)x^2+(6a+4)x+5a$
- $y^2=4ax^6+5x^5+(4a+3)x^4+5x^3+(4a+1)x^2+(a+5)x+6a+4$
- $y^2=(5a+4)x^6+(5a+5)x^5+(2a+3)x^4+5ax^3+3x^2+(4a+6)x+a+1$
- $y^2=(3a+3)x^6+3x^5+x^3+x^2+(3a+2)x+6a+6$
- $y^2=(6a+2)x^6+(a+2)x^4+(a+4)x^3+(a+4)x^2+(a+3)x+2a+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.14725.1. |
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 |
---|---|---|
2.49.v_hp | $2$ | (not in LMFDB) |