Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 22 x + 216 x^{2} - 1078 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.136516647719$, $\pm0.269709412854$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.537408.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1518$ | $5643924$ | $13885414686$ | $33263775484368$ | $79801769045878638$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $28$ | $2350$ | $118024$ | $5770150$ | $282508888$ | $13841400094$ | $678223157836$ | $33232931475838$ | $1628413640423404$ | $79792266857711230$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(6a+2)x^6+(6a+4)x^5+(3a+1)x^4+2x^3+3ax^2+(3a+2)x+3a$
- $y^2=(5a+5)x^6+(4a+5)x^5+(a+4)x^4+(4a+5)x^3+(3a+1)x^2+4ax+5a+1$
- $y^2=(3a+6)x^6+2x^5+3ax^4+(3a+4)x^3+(3a+2)x^2+(5a+2)x+5a+1$
- $y^2=(4a+3)x^6+4x^5+6x^4+(3a+3)x^3+(5a+1)x+6a+6$
- $y^2=(a+5)x^6+ax^5+(3a+6)x^4+(5a+3)x^3+(2a+5)x^2+(3a+5)x+a+4$
- $y^2=(5a+4)x^6+4ax^5+(5a+6)x^4+(5a+3)x^3+(3a+5)x^2+5ax+3$
- $y^2=(4a+5)x^6+6x^5+5x^4+(5a+5)x^3+(3a+5)x^2+(6a+3)x+6a+2$
- $y^2=(a+4)x^6+(a+6)x^5+(4a+4)x^4+2x^3+(2a+1)x^2+(2a+3)x+6a+1$
- $y^2=(5a+5)x^6+x^5+5ax^4+6x^3+(a+3)x^2+(4a+5)x+2a+2$
- $y^2=(2a+2)x^6+(5a+5)x^5+(a+4)x^4+(5a+2)x^3+(a+5)x^2+(2a+3)x+3a+5$
- $y^2=(2a+5)x^6+(4a+1)x^5+(6a+5)x^4+(3a+1)x^3+(6a+6)x^2+(2a+5)x+a+1$
- $y^2=5ax^6+(5a+5)x^5+(4a+4)x^4+(4a+3)x^3+4x^2+(a+5)x+6a$
- $y^2=5ax^6+(a+3)x^5+(6a+3)x^4+3x^3+(5a+1)x^2+(3a+5)x+5a+1$
- $y^2=(5a+3)x^6+(2a+4)x^5+(a+1)x^4+(5a+1)x^3+(2a+6)x^2+(2a+6)x+a$
- $y^2=(5a+5)x^6+6x^5+(a+3)x^4+(3a+3)x^3+(4a+2)x^2+(5a+5)x+a+5$
- $y^2=6ax^6+(5a+5)x^5+4x^4+(2a+3)x^3+5ax^2+6ax+5a$
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 4.0.537408.1. |
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.w_ii | $2$ | (not in LMFDB) |