Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 21 x + 198 x^{2} - 1029 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.0658419374597$, $\pm0.325440921041$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1983580.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1550$ | $5657500$ | $13855964600$ | $33230344600000$ | $79783719103742750$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $29$ | $2357$ | $117776$ | $5764353$ | $282444989$ | $13840978322$ | $678221813501$ | $33232935220993$ | $1628413699759904$ | $79792266961225877$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(3a+5)x^6+5x^5+(4a+6)x^4+(6a+6)x^3+(5a+1)x^2+(4a+3)x+4a+3$
- $y^2=3ax^6+(a+3)x^5+5ax^4+4x^3+2ax^2+(6a+1)x+a+3$
- $y^2=2ax^6+(3a+1)x^5+(3a+2)x^4+(a+3)x^3+(5a+3)x^2+(a+2)x+5a+3$
- $y^2=5x^6+(2a+2)x^5+(6a+3)x^4+(4a+4)x^3+(5a+1)x^2+2ax+5a+5$
- $y^2=4ax^6+3x^5+3x^4+(6a+6)x^3+(5a+6)x^2+(2a+6)x+4a+6$
- $y^2=(a+5)x^6+6ax^5+(2a+5)x^4+(5a+5)x^3+6ax^2+(6a+5)x+4a$
- $y^2=2ax^6+3x^5+(2a+2)x^4+(6a+2)x^3+(3a+2)x^2+(2a+5)x+4a+3$
- $y^2=(3a+4)x^6+2ax^5+(5a+5)x^4+(4a+5)x^3+6x^2+(5a+2)x+6a+3$
- $y^2=(4a+2)x^6+(a+2)x^5+(2a+4)x^4+(2a+4)x^3+2x^2+(6a+4)x+1$
- $y^2=(5a+2)x^6+(4a+4)x^5+(6a+3)x^4+(5a+1)x^3+(5a+1)x^2+(6a+4)x+6a+4$
- $y^2=(4a+6)x^6+(6a+2)x^5+(6a+5)x^3+ax^2+(2a+1)x+5a+2$
- $y^2=2ax^6+6ax^5+(a+5)x^4+x^3+(a+3)x^2+(4a+1)x+3a+4$
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 4.0.1983580.1. |
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_hq | $2$ | (not in LMFDB) |