Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 20 x + 191 x^{2} - 980 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.141161076953$, $\pm0.323951802917$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.446096.4 |
Galois group: | $D_{4}$ |
Jacobians: | $18$ |
Isomorphism classes: | 18 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1593$ | $5723649$ | $13902588900$ | $33255093251529$ | $79794498166934553$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $30$ | $2384$ | $118170$ | $5768644$ | $282483150$ | $13841264198$ | $678223620510$ | $33232944433924$ | $1628413729082010$ | $79792266864897104$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+6)x^6+x^5+(2a+5)x^4+6ax^3+ax^2+2ax+3a+1$
- $y^2=(2a+2)x^6+2ax^5+2x^4+(2a+3)x^3+2x^2+3x+5a+1$
- $y^2=(6a+2)x^6+(5a+6)x^5+(a+3)x^4+(5a+3)x^3+(4a+4)x^2+(5a+2)x+a+6$
- $y^2=(6a+5)x^6+5ax^5+(2a+1)x^4+(2a+1)x^3+3ax^2+(3a+1)x+3$
- $y^2=4ax^6+(6a+6)x^5+(4a+5)x^4+(6a+5)x^3+(a+4)x^2+(a+4)x+a+2$
- $y^2=(3a+4)x^6+(3a+2)x^5+(6a+6)x^3+(5a+5)x^2+(5a+6)x+2a+2$
- $y^2=(4a+3)x^6+6ax^5+(a+2)x^4+(2a+1)x^3+ax^2+5ax+2a+3$
- $y^2=(3a+5)x^6+(2a+4)x^5+3x^4+(6a+5)x^3+(a+4)x^2+(6a+4)x+4a+3$
- $y^2=3x^6+2ax^5+(5a+6)x^4+(6a+1)x^3+3ax^2+(a+3)x+3a+1$
- $y^2=(3a+3)x^6+5x^5+(3a+6)x^4+(a+3)x^3+2x^2+5a+4$
- $y^2=2ax^6+(a+6)x^5+6x^4+(6a+4)x^3+(5a+1)x^2+(2a+1)x+2a+3$
- $y^2=(5a+2)x^6+(3a+5)x^5+6ax^4+(a+6)x^3+2x^2+6x+3a+3$
- $y^2=(6a+2)x^6+(5a+4)x^5+(5a+6)x^4+(4a+3)x^3+(2a+1)x^2+2x+a+3$
- $y^2=(4a+4)x^6+(3a+6)x^5+(a+5)x^4+(6a+2)x^3+6ax^2+(6a+1)x+4a+4$
- $y^2=6ax^6+5x^5+(a+5)x^4+3x^3+(4a+6)x^2+(a+1)x+a+6$
- $y^2=(2a+4)x^6+x^5+(2a+5)x^4+(2a+5)x^3+(5a+2)x^2+(a+3)x+4a+5$
- $y^2=ax^6+(2a+5)x^5+(4a+3)x^4+(5a+1)x^3+(2a+3)x^2+(4a+5)x+6a+2$
- $y^2=ax^6+(4a+2)x^5+(5a+5)x^4+(4a+3)x^3+(a+5)x+4$
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 4.0.446096.4. |
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_hj | $2$ | (not in LMFDB) |