Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 25 x^{2} - 77 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.0530380253560$, $\pm0.477974681599$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.72557.1 |
Galois group: | $D_{4}$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $63$ | $14553$ | $1707993$ | $208238877$ | $25807119408$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $123$ | $1283$ | $14219$ | $160240$ | $1772559$ | $19487809$ | $214327075$ | $2357884139$ | $25937703318$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+10x^5+3x^4+4x^3+7x^2+7x$
- $y^2=3x^5+6x^4+2x+6$
- $y^2=7x^6+6x^5+7x^4+9x^3+4x^2+4x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$The endomorphism algebra of this simple isogeny class is 4.0.72557.1. |
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 |
---|---|---|
2.11.h_z | $2$ | 2.121.b_aid |