Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 20 x + 188 x^{2} - 980 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.110672799774$, $\pm0.337577521471$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.5433600.1 |
Galois group: | $D_{4}$ |
Jacobians: | $16$ |
Isomorphism classes: | 32 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1590$ | $5708100$ | $13881286710$ | $33240526707600$ | $79788566331849750$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $30$ | $2378$ | $117990$ | $5766118$ | $282462150$ | $13841168618$ | $678223637310$ | $33232948154878$ | $1628413758738270$ | $79792267046027498$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+1)x^6+(2a+5)x^5+x^4+(6a+1)x^3+(a+1)x^2+(4a+3)x+a+1$
- $y^2=3x^6+(4a+1)x^5+(6a+6)x^4+(3a+3)x^3+2ax^2+(a+1)x+4a+4$
- $y^2=(4a+1)x^6+(6a+2)x^5+(6a+2)x^4+(a+5)x^3+(3a+3)x^2+(5a+4)x+2a+2$
- $y^2=(a+5)x^6+ax^5+(4a+5)x^4+6ax^3+ax^2+(3a+6)x$
- $y^2=(5a+6)x^6+2ax^5+(5a+2)x^4+(6a+2)x^3+5x^2+3ax+2a+6$
- $y^2=(3a+1)x^6+(5a+2)x^5+(4a+5)x^4+(2a+1)x^3+(4a+2)x^2+3x+6a+1$
- $y^2=(4a+1)x^6+(2a+1)x^5+(a+3)x^4+6x^3+(5a+2)x^2+3x+4a+1$
- $y^2=(3a+3)x^6+6ax^5+(4a+3)x^4+5ax^3+(6a+3)x^2+(4a+5)x+2a+4$
- $y^2=(3a+2)x^6+(2a+5)x^5+(6a+5)x^4+5ax^3+(5a+2)x^2+(6a+3)x+6$
- $y^2=(a+1)x^6+(3a+6)x^5+(6a+2)x^4+(a+6)x^3+3ax^2+x+4a+1$
- $y^2=(5a+4)x^6+(3a+3)x^5+(2a+5)x^4+(5a+5)x^3+(3a+2)x^2+(2a+4)x+5a$
- $y^2=6ax^6+2x^4+(2a+5)x^3+(2a+4)x^2+(3a+2)x+5a$
- $y^2=(5a+4)x^6+(a+5)x^5+(a+5)x^4+2x^3+(4a+1)x^2+(6a+5)x+a+5$
- $y^2=(6a+1)x^6+(a+6)x^5+2x^4+5x^3+(a+2)x^2+(5a+2)x+a+6$
- $y^2=(5a+1)x^5+(2a+4)x^4+2ax^3+(a+3)x^2+(2a+1)x+2a$
- $y^2=(5a+3)x^6+(a+3)x^5+(2a+1)x^4+(4a+3)x^3+(2a+4)x^2+3ax+3a+1$
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 4.0.5433600.1. |
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_hg | $2$ | (not in LMFDB) |