Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 21 x + 199 x^{2} - 1029 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.0816995674884$, $\pm0.321155419757$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2427237.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1551$ | $5662701$ | $13863422727$ | $33235938086373$ | $79786477354344816$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $29$ | $2359$ | $117839$ | $5765323$ | $282454754$ | $13841050831$ | $678222270083$ | $33232938408019$ | $1628413728252977$ | $79792267216439614$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(5a+2)x^6+(a+1)x^5+(a+6)x^4+ax^3+ax^2+(3a+5)x+a$
- $y^2=(5a+2)x^6+(2a+5)x^5+(6a+6)x^4+(3a+1)x^3+6x^2+(5a+1)x+5a+5$
- $y^2=(4a+4)x^6+2ax^5+4x^4+3x^3+(6a+3)x^2+(a+2)x+2a+5$
- $y^2=6ax^6+(4a+2)x^5+x^4+(a+2)x^3+(3a+5)x^2+(5a+4)x$
- $y^2=(3a+3)x^6+(2a+3)x^5+(3a+4)x^4+(3a+5)x^3+(a+5)x^2+5ax+4a+4$
- $y^2=(a+1)x^6+(6a+1)x^5+(5a+6)x^4+4ax^3+(a+6)x^2+(a+6)x+3a+3$
- $y^2=(4a+1)x^6+(5a+4)x^5+6ax^4+4x^3+2ax^2+x+a+4$
- $y^2=(6a+1)x^6+(2a+1)x^5+2ax^4+(4a+2)x^3+5x^2+(5a+6)x+4a+4$
- $y^2=6ax^6+2ax^5+(6a+5)x^4+(a+4)x^3+(6a+3)x^2+(3a+3)x+1$
- $y^2=(4a+4)x^5+(6a+1)x^4+(6a+4)x^3+5ax^2+(5a+5)x+a+5$
- $y^2=6ax^6+(a+4)x^5+2x^4+(6a+6)x^3+6x^2+(a+2)x+6$
- $y^2=2ax^6+4x^5+6x^4+5x^3+(6a+4)x^2+(6a+1)x+a$
- $y^2=3x^6+x^4+(4a+6)x^3+(5a+1)x^2+3x+a$
- $y^2=3ax^6+(3a+2)x^5+(4a+5)x^4+x^3+(3a+4)x^2+(2a+5)x+a+1$
- $y^2=(a+1)x^6+(2a+2)x^5+(3a+3)x^4+(2a+4)x^3+6ax^2+(4a+3)x+6a+2$
- $y^2=5x^6+(4a+1)x^5+(a+2)x^4+3x^3+(5a+4)x^2+(a+1)x+2a+5$
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 4.0.2427237.1. |
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.v_hr | $2$ | (not in LMFDB) |