Invariants
Base field: | $\F_{11^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 37 x + 583 x^{2} - 4477 x^{3} + 14641 x^{4}$ |
Frobenius angles: | $\pm0.149492779409$, $\pm0.210034091690$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10711$ | $211424429$ | $3139545778471$ | $45957133832679749$ | $672764106260287923376$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $85$ | $14439$ | $1772191$ | $214393419$ | $25937968650$ | $3138434159583$ | $379749875782075$ | $45949729983567699$ | $5559917311254608641$ | $672749994884067873454$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+4)x^6+(5a+2)x^5+(9a+3)x^4+(2a+7)x^3+(3a+5)x^2+(10a+2)x+2a+2$
- $y^2=ax^6+5ax^5+(3a+6)x^4+(10a+4)x^3+7x^2+(8a+10)x+8a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.3725.1. |
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 |
---|---|---|
2.121.bl_wl | $2$ | (not in LMFDB) |