Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 8 x^{2} - 28 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.0704914820143$, $\pm0.570491482014$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(i, \sqrt{10})\) |
Galois group: | $C_2^2$ |
Jacobians: | $3$ |
Isomorphism classes: | 6 |
This isogeny class is simple but not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $26$ | $2340$ | $101114$ | $5475600$ | $284431706$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $50$ | $292$ | $2278$ | $16924$ | $117650$ | $821692$ | $5766718$ | $40370404$ | $282475250$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+6x^5+x^4+2x^3+6x^2+x$
- $y^2=x^6+4x^5+5x^4+5x^2+3x+1$
- $y^2=2x^5+2x^4+2x^3+6x^2+4x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{4}}$.
Endomorphism algebra over $\F_{7}$The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{10})\). |
The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ack 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$ |
- Endomorphism algebra over $\F_{7^{2}}$
The base change of $A$ to $\F_{7^{2}}$ is the simple isogeny class 2.49.a_ack and its endomorphism algebra is \(\Q(i, \sqrt{10})\).
Base change
This is a primitive isogeny class.