Invariants
Base field: | $\F_{11^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 39 x + 617 x^{2} - 4719 x^{3} + 14641 x^{4}$ |
Frobenius angles: | $\pm0.0438805027714$, $\pm0.214090256206$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.757197.2 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10501$ | $210198517$ | $3136149114925$ | $45950031666713925$ | $672751698372621561856$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $83$ | $14355$ | $1770275$ | $214360291$ | $25937490278$ | $3138428197347$ | $379749811557563$ | $45949729404540931$ | $5559917307371824475$ | $672749994875773455630$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(8a+8)x^6+(10a+10)x^5+(4a+8)x^4+3x^3+(5a+10)x^2+(6a+5)x+8a+10$
- $y^2=(2a+9)x^6+(4a+5)x^5+4x^4+(6a+4)x^2+(9a+5)x+9a+6$
- $y^2=(9a+8)x^6+(8a+5)x^5+(5a+3)x^4+(8a+3)x^3+(8a+5)x^2+(4a+10)x+5a+2$
- $y^2=(10a+5)x^6+(4a+7)x^5+(7a+5)x^4+(4a+8)x^2+(7a+6)x+6a+3$
- $y^2=(9a+6)x^6+(3a+4)x^5+(10a+9)x^4+(3a+10)x^3+(10a+7)x^2+(10a+10)x+4a+2$
- $y^2=6ax^6+(2a+7)x^5+(4a+3)x^4+(a+4)x^3+(3a+6)x^2+(10a+5)x+2a+3$
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 4.0.757197.2. |
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.bn_xt | $2$ | (not in LMFDB) |