Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 20 x + 184 x^{2} - 980 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.0612448027520$, $\pm0.352483684171$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3524864.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1586$ | $5687396$ | $13852899554$ | $33220785273104$ | $79779866673449826$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $30$ | $2370$ | $117750$ | $5762694$ | $282431350$ | $13840971906$ | $678222538590$ | $33232939938174$ | $1628413679999550$ | $79792266449027650$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(6a+2)x^6+(5a+5)x^5+(a+3)x^4+(5a+4)x^3+(4a+1)x^2+2ax+2a+5$
- $y^2=(5a+2)x^6+5ax^5+(a+4)x^4+(4a+2)x^3+(3a+4)x^2+(4a+3)x+2a$
- $y^2=(6a+2)x^6+(4a+6)x^5+(a+5)x^4+6ax^3+2x^2+(3a+3)x+3a+3$
- $y^2=(5a+2)x^6+3x^5+(a+6)x^4+(3a+6)x^3+(5a+3)x^2+(6a+3)x+4a+1$
- $y^2=(a+1)x^6+6ax^5+(6a+4)x^4+3x^3+(3a+6)x^2+(2a+5)x+2a+3$
- $y^2=(a+5)x^6+(2a+4)x^5+4ax^4+(3a+1)x^3+(3a+4)x^2+(5a+4)x+2a+5$
- $y^2=(2a+5)x^6+6ax^5+(4a+1)x^4+ax^3+(5a+2)x^2+x+4a+5$
- $y^2=(a+4)x^6+(a+5)x^5+5x^4+(5a+5)x^3+(2a+4)x^2+(5a+5)x+2a+5$
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 4.0.3524864.2. |
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.u_hc | $2$ | (not in LMFDB) |