Invariants
Base field: | $\F_{2^{7}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 37 x + 597 x^{2} - 4736 x^{3} + 16384 x^{4}$ |
Frobenius angles: | $\pm0.166043376882$, $\pm0.221161776642$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.667025.1 |
Galois group: | $D_{4}$ |
Jacobians: | $35$ |
Isomorphism classes: | 35 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12209$ | $265606795$ | $4400999278316$ | $72070158117798275$ | $1180614630343474885669$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $92$ | $16210$ | $2098559$ | $268482258$ | $34360408032$ | $4398052874455$ | $562949987690084$ | $72057593911688898$ | $9223372030890542087$ | $1180591620630139135050$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 35 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^6+a^4+a^2+1)x+a^6+a^4+a^2+1)y=x^6+(a^6+a^4+a^3+a+1)x^5+(a^6+a^4+a^3+a+1)x^4+(a^6+a^4+1)x^3+(a^6+a^5+a^4+a^2+a+1)x^2+(a^6+a^3+a)x+a^4+a^2+a$
- $y^2+(x^3+(a^5+a^3+1)x+a^5+a^3+1)y=x^6+a^2x^5+a^2x^4+(a^3+a^2)x^3+(a^6+a^4+a^3+1)x^2+(a^5+a^3+a^2)x+a^6+a^4+a$
- $y^2+(x^3+(a^6+a^2+1)x+a^6+a^2+1)y=x^6+a^4x^5+a^4x^4+(a^6+a^4+a^3)x^3+(a^6+a^5+a^4+1)x^2+a^5+a^4+a^3+1$
- $y^2+(x^3+(a^5+a^3+1)x+a^5+a^3+1)y=(a^6+a^4+a^2+a)x^5+(a^6+a^4+a^2+a)x^4+(a^5+a^4+1)x^3+(a^6+a^5+a^4+a^2)x+a^6+a^5+a$
- $y^2+(x^3+(a^6+a^5+a^2+a+1)x+a^6+a^5+a^2+a+1)y=x^6+(a^4+a)x^5+(a^4+a)x^4+(a^6+a^4+a^3)x^3+(a^6+a^5+a^3+a^2+1)x^2+(a^6+a^5+a^4+a^2)x+a^5+a^4$
- $y^2+(x^3+(a^5+a^4+a^2+1)x+a^5+a^4+a^2+1)y=x^6+(a^6+a^5+a^4+a^3+a+1)x^5+(a^6+a^5+a^4+a^3+a+1)x^4+(a^5+a^2+a+1)x^3+(a^3+a^2+a+1)x^2+(a^6+a^5+a^4+a^3)x+a^2+a$
- $y^2+(x^3+(a^5+a^4+a^3+a^2+1)x+a^5+a^4+a^3+a^2+1)y=x^6+(a^6+a^4+a^2+a+1)x^5+(a^6+a^4+a^2+a+1)x^4+(a^5+a^3+1)x^3+(a^6+a^3+a^2+a+1)x^2+a^6x+a^2$
- $y^2+(x^3+(a^3+a^2+1)x+a^3+a^2+1)y=x^6+(a^4+a^2)x^5+(a^4+a^2)x^4+(a^6+a^5+a^3+a^2)x^3+(a^6+a^4+1)x^2+(a^4+a^3)x+a^5+a^3+a^2+a$
- $y^2+(x^3+(a^6+a^5+a^4+a^2+a+1)x+a^6+a^5+a^4+a^2+a+1)y=x^6+(a^5+a+1)x^5+(a^5+a+1)x^4+(a^6+a^5+a^2+a+1)x^3+(a^6+a^5+a^3+a+1)x^2+(a^5+a^2)x+a^4+a$
- $y^2+(x^3+(a^6+a^4+a^3+1)x+a^6+a^4+a^3+1)y=x^6+a^4x^5+a^4x^4+(a^6+a^4)x^3+(a^5+a^2+a+1)x^2+(a^6+a^3)x+a^6+a^5+a$
- $y^2+(x^3+(a^5+a^3+a+1)x+a^5+a^3+a+1)y=(a^5+a^4+a+1)x^5+(a^5+a^4+a+1)x^4+a^4x^3+(a^5+a^4+a+1)x+a^3+a^2+a+1$
- $y^2+(x^3+(a^5+a^4+a^2+a+1)x+a^5+a^4+a^2+a+1)y=(a^6+a+1)x^5+(a^6+a+1)x^4+(a^4+a^2)x^3+(a^6+a+1)x+a^6+a^5+a^4+a^2+a+1$
- $y^2+(x^3+(a^6+a^4+a^3+a^2+1)x+a^6+a^4+a^3+a^2+1)y=x^6+(a^4+a)x^5+(a^4+a)x^4+(a^6+a^5+a^2+a)x^3+(a^5+a^4+a^2+a+1)x^2+a^6+a+1$
- $y^2+(x^3+(a^6+a^2+1)x+a^6+a^2+1)y=(a^6+a^5+a^3+1)x^5+(a^6+a^5+a^3+1)x^4+(a^4+a)x^3+(a^6+a^5+a^3+1)x+a^5+a^4+a^3+a^2+1$
- $y^2+(x^3+(a^6+a^5+a^4+1)x+a^6+a^5+a^4+1)y=x^6+(a^2+a)x^5+(a^2+a)x^4+(a^5+a^2+a)x^3+(a^6+a^5+a^4+a^3+a^2+a+1)x^2+a^6+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a^6+a^4+a^3+1)x+a^6+a^4+a^3+1)y=(a^6+a^5+a^4+a)x^5+(a^6+a^5+a^4+a)x^4+(a^4+a^3+a^2+a+1)x^3+(a^6+a^5+a^3+a^2+a)x+a^6+a^5+a^4+a^3+a^2$
- $y^2+(x^3+(a^3+a^2+a+1)x+a^3+a^2+a+1)y=x^6+(a^5+a^4+a^3+a+1)x^5+(a^5+a^4+a^3+a+1)x^4+(a^3+a^2+1)x^3+(a^6+a^4+a^2+1)x^2+(a^5+a^4+a^3+a^2+a)x+a^4+a^2$
- $y^2+(x^3+(a^6+a^5+a^3+a^2+1)x+a^6+a^5+a^3+a^2+1)y=x^6+ax^5+ax^4+(a^5+a^2+a)x^3+(a^5+a^3+1)x^2+(a^6+a^5+a^3+a^2+a)x+a^4+a^3+a^2+a$
- $y^2+(x^3+(a^3+a+1)x+a^3+a+1)y=x^6+a^2x^5+a^2x^4+(a^5+a^3)x^3+(a^6+a^2+1)x^2+a^6+a^5+a^3+1$
- $y^2+(x^3+(a^6+a^5+a^3+a^2+1)x+a^6+a^5+a^3+a^2+1)y=(a^4+a^3+a^2)x^5+(a^4+a^3+a^2)x^4+(a^6+a^3+a^2+1)x^3+(a^6+a^2+a)x+a^6+a^4+a$
- and 15 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{7}}$.
Endomorphism algebra over $\F_{2^{7}}$The endomorphism algebra of this simple isogeny class is 4.0.667025.1. |
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 |
---|---|---|
2.128.bl_wz | $2$ | (not in LMFDB) |