Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 7 x + 16 x^{2} )( 1 - 3 x + 16 x^{2} )$ |
| $1 - 10 x + 53 x^{2} - 160 x^{3} + 256 x^{4}$ | |
| Frobenius angles: | $\pm0.160861246510$, $\pm0.377642706461$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $16$ |
| Isomorphism classes: | 80 |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $140$ | $67200$ | $17235260$ | $4308595200$ | $1099248363500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $263$ | $4207$ | $65743$ | $1048327$ | $16779863$ | $268479967$ | $4295179423$ | $68719726327$ | $1099509325223$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=(a^2+1)x^5+(a^3+a^2+a+1)x^4+(a^2+1)x^3+x^2+(a^3+a^2+a)x$
- $y^2+(x^2+x)y=(a+1)x^5+(a^3+a^2+1)x^4+(a+1)x^3+(a^3+a^2+1)x$
- $y^2+(x^2+x)y=a^2x^5+(a^3+1)x^4+a^2x^3+(a^3+1)x$
- $y^2+(x^2+x)y=ax^5+a^3x^4+ax^3+(a+1)x^2+(a^3+a+1)x$
- $y^2+(x^2+x+a^3+a+1)y=(a^2+a+1)x^5+a^3x^4+(a^3+a^2)x^3+a^3x^2+a^3x+a^3+a^2+1$
- $y^2+(x^2+x+a^3)y=x^5+a^3x^4+a^3x^3+a^3x^2+x+a^3+a^2+a$
- $y^2+(x^2+x+a^3+1)y=(a^2+a)x^5+(a^3+a+1)x^4+(a^3+a^2+a+1)x^3+(a^3+a+1)x^2+(a^3+a^2)x+a^3+a+1$
- $y^2+(x^2+x+a^3+a^2)y=x^5+(a^3+a)x^4+(a^3+a^2)x^3+(a^3+a)x^2+x+a^3+a^2+a+1$
- $y^2+(x^2+x+a^3+a^2+1)y=(a^2+a+1)x^5+(a^3+a+1)x^4+(a^3+a)x^3+(a^3+a+1)x^2+(a^3+a^2+a+1)x+a^2+a$
- $y^2+(x^2+x+a^3+a^2+a+1)y=x^5+(a^3+a^2+a+1)x^4+(a^3+a^2+a+1)x^3+(a^3+a^2+a+1)x^2+x+a^3+1$
- $y^2+(x^2+x)y=(a+1)x^5+a^3x^4+(a+1)x^3+ax^2+(a^3+a)x$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^2+a)x^5+(a^3+a)x^4+a^3x^3+(a^3+a)x^2+(a^3+a)x+a^3+1$
- $y^2+(x^2+x+a^3+a)y=x^5+a^3x^4+(a^3+a)x^3+a^3x^2+x+a^3+a^2+a$
- $y^2+(x^2+x)y=a^2x^5+(a^3+1)x^4+a^2x^3+(a^2+a)x^2+(a^3+a^2+a+1)x$
- $y^2+(x^2+x)y=ax^5+a^3x^4+ax^3+a^2x^2+(a^3+a^2)x$
- $y^2+(x^2+x)y=(a^2+1)x^5+(a^3+a^2+1)x^4+(a^2+1)x^3+(a^2+1)x^2+a^3x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$| The isogeny class factors as 1.16.ah $\times$ 1.16.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ae_l | $2$ | 2.256.g_er |
| 2.16.e_l | $2$ | 2.256.g_er |
| 2.16.k_cb | $2$ | 2.256.g_er |