Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 20 x + 192 x^{2} - 980 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.151227437545$, $\pm0.318680514787$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3283200.1 |
Galois group: | $D_{4}$ |
Jacobians: | $16$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1594$ | $5728836$ | $13909691914$ | $33259903165200$ | $79796362518183754$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $30$ | $2386$ | $118230$ | $5769478$ | $282489750$ | $13841286226$ | $678223459230$ | $33232941467518$ | $1628413705912830$ | $79792266746255506$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(5a+4)x^6+(3a+3)x^5+(3a+5)x^4+(3a+4)x^3+x^2+(4a+2)x+5a+2$
- $y^2=(3a+3)x^6+(3a+5)x^5+(6a+1)x^4+(3a+5)x^3+x^2+6x+a$
- $y^2=(6a+6)x^6+(2a+4)x^5+3ax^4+(3a+4)x^3+(a+5)x^2+6ax+2a+4$
- $y^2=(2a+1)x^6+(3a+3)x^5+(2a+5)x^4+(6a+6)x^3+6x^2+6x+6a+2$
- $y^2=(3a+2)x^6+ax^5+(5a+3)x^4+(5a+1)x^3+6ax^2+(5a+2)x+a+2$
- $y^2=(4a+2)x^6+(4a+1)x^5+(3a+3)x^3+(6a+5)x^2+(5a+2)x+1$
- $y^2=(4a+1)x^6+(6a+5)x^5+(a+1)x^3+5x^2+(6a+6)x+3a+5$
- $y^2=2ax^6+5ax^5+(3a+4)x^4+4ax^3+(6a+6)x+2a+2$
- $y^2=(6a+4)x^6+(2a+2)x^5+(2a+3)x^4+(2a+1)x^3+(6a+3)x^2+(a+6)x+4a+4$
- $y^2=(2a+2)x^6+(2a+2)x^5+2ax^4+(4a+2)x^3+(5a+3)x^2+(6a+5)x+4a+1$
- $y^2=6ax^6+(3a+5)x^5+(5a+4)x^3+(3a+4)x^2+(6a+6)x+6a+1$
- $y^2=5x^6+(2a+6)x^5+(4a+4)x^3+(3a+2)x^2+(a+5)x+5a+6$
- $y^2=(3a+4)x^6+(2a+5)x^5+(3a+6)x^4+4ax^3+(6a+6)x^2+(4a+3)x+6a+5$
- $y^2=(6a+4)x^6+(5a+3)x^5+(3a+2)x^4+(4a+4)x^3+(4a+1)x^2+(6a+5)x+4$
- $y^2=(4a+3)x^6+(4a+3)x^5+2ax^4+(5a+4)x^3+(4a+2)x^2+(a+3)x+6a+6$
- $y^2=2x^6+(3a+2)x^5+(3a+5)x^4+x^3+(2a+3)x^2+3ax+5a+4$
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 4.0.3283200.1. |
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.u_hk | $2$ | (not in LMFDB) |