Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 23 x + 227 x^{2} - 1127 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.100880320937$, $\pm0.256439151204$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.328029.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1479$ | $5589141$ | $13854847527$ | $33252264620181$ | $79798399975066224$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $27$ | $2327$ | $117765$ | $5768155$ | $282496962$ | $13841335127$ | $678222749217$ | $33232928665939$ | $1628413629471855$ | $79792266975853502$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+5)x^6+(2a+5)x^5+(6a+5)x^4+(5a+4)x^3+6ax^2+4x+6a+3$
- $y^2=(6a+2)x^6+(4a+3)x^5+3ax^3+(5a+3)x^2+x+6a$
- $y^2=(4a+3)x^6+2ax^5+(3a+4)x^3+(4a+1)x^2+(a+5)x+a+1$
- $y^2=(a+4)x^6+(2a+4)x^5+(5a+4)x^4+(5a+3)x^3+(a+1)x^2+(3a+4)x+6a+6$
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 4.0.328029.1. |
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.x_it | $2$ | (not in LMFDB) |