Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 39 x^{2} - 112 x^{3} + 256 x^{4}$ |
Frobenius angles: | $\pm0.242342613760$, $\pm0.451721988494$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.840105.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $177$ | $73455$ | $17363700$ | $4300716795$ | $1099724316387$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $286$ | $4237$ | $65626$ | $1048780$ | $16783423$ | $268445110$ | $4294792786$ | $68718521797$ | $1099511241526$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+1)x^6+x^5+x^4+(a^2+a)x^3+a^2x^2+(a^3+a^2)x+a^3+a^2$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a^2)x^6+(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+(a^3+a)x^3+(a^2+a+1)x^2+(a^2+1)x+a^3+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a)x^6+a^3x^5+a^3x^4+(a^3+a^2)x^3+(a^2+a+1)x^2+a^2x+a^3+a^2+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2+a)x^6+x^5+x^4+(a^2+a+1)x^3+(a+1)x^2+(a^3+a^2+a+1)x+a^3+a^2+a+1$
- $y^2+(x^3+ax+a)y=(a^3+1)x^6+x^5+x^4+(a^2+a+1)x^3+ax^2+a^3x+a^3$
- $y^2+(x^3+ax+a)y=(a^3+a^2+1)x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^3+a^2+a+1)x^3+x^2+(a+1)x+a^3+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2+a)x^6+(a^3+a)x^5+(a^3+a)x^4+a^3x^3+(a+1)x^2+ax+a^3+a^2$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2+a+1)x^6+x^5+x^4+(a^2+a)x^3+(a^3+a^2+1)x^2+(a^3+a)x+a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$The endomorphism algebra of this simple isogeny class is 4.0.840105.1. |
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 |
---|---|---|
2.16.h_bn | $2$ | 2.256.bd_rx |