Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 21 x + 207 x^{2} - 1029 x^{3} + 2401 x^{4}$ |
Frobenius angles: | $\pm0.188420471035$, $\pm0.266233865926$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.164725.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1559$ | $5704381$ | $13923131231$ | $33279866443909$ | $79806409265639024$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $29$ | $2375$ | $118343$ | $5772939$ | $282525314$ | $13841432111$ | $678222461771$ | $33232919719699$ | $1628413525463537$ | $79792266045429950$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+4)x^6+2ax^5+(4a+4)x^4+(2a+5)x^3+(6a+4)x^2+(2a+2)x+4a+6$
- $y^2=(6a+6)x^6+(3a+1)x^5+(4a+5)x^4+(4a+2)x^3+3ax^2+(4a+3)x$
- $y^2=(5a+2)x^6+(3a+1)x^5+(5a+3)x^4+(4a+2)x^3+3x^2+5x+5a+5$
- $y^2=(2a+2)x^6+(4a+5)x^5+(4a+3)x^4+(6a+2)x^3+(2a+5)x^2+(4a+1)x+a+3$
- $y^2=(a+5)x^6+(4a+4)x^5+(3a+2)x^4+(3a+6)x^3+(4a+3)x^2+3ax$
- $y^2=(5a+4)x^6+(2a+3)x^5+(a+5)x^4+(3a+6)x^3+(5a+2)x^2+(2a+1)x+5a+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.164725.1. |
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 |
---|---|---|
2.49.v_hz | $2$ | (not in LMFDB) |