Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 22 x + 212 x^{2} - 1078 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.0717588176879$, $\pm0.296465529446$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.969024.6 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1514$ | $5622996$ | $13854118922$ | $33238856383056$ | $79788718381561274$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $28$ | $2342$ | $117760$ | $5765830$ | $282462688$ | $13841055542$ | $678221483548$ | $33232928706814$ | $1628413674293020$ | $79792267215887702$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3ax^6+(3a+5)x^5+(6a+6)x^4+(4a+5)x^3+(3a+4)x^2+ax+6a+1$
- $y^2=(4a+4)x^6+(a+2)x^5+(6a+2)x^4+(a+5)x^3+(3a+2)x^2+(5a+2)x+4a$
- $y^2=ax^6+(4a+1)x^5+(2a+6)x^4+(4a+3)x^3+(a+3)x^2+x+3a$
- $y^2=(6a+5)x^6+6ax^5+(3a+5)x^4+(2a+6)x^3+(a+4)x+a+5$
- $y^2=(5a+5)x^6+(2a+6)x^5+(4a+4)x^4+(6a+6)x^3+(a+4)x+5a+5$
- $y^2=(a+6)x^6+(6a+4)x^5+4x^4+(a+4)x^3+(3a+2)x^2+(5a+3)x+4a+4$
- $y^2=(5a+1)x^6+(a+2)x^5+(3a+6)x^4+5x^3+(3a+6)x^2+3x+5$
- $y^2=2ax^6+ax^5+(3a+2)x^4+(5a+1)x^3+(3a+4)x^2+(3a+4)x+4a+3$
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 4.0.969024.6. |
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.w_ie | $2$ | (not in LMFDB) |