Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 12 x + 49 x^{2} )( 1 - 9 x + 49 x^{2} )$ |
$1 - 21 x + 206 x^{2} - 1029 x^{3} + 2401 x^{4}$ | |
Frobenius angles: | $\pm0.172237328522$, $\pm0.277748883973$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 20 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1558$ | $5699164$ | $13915663384$ | $33274455030720$ | $79804125241799398$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $29$ | $2373$ | $118280$ | $5772001$ | $282517229$ | $13841403666$ | $678222766061$ | $33232926047041$ | $1628413585185080$ | $79792266366710853$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+2)x^6+(a+2)x^5+(a+6)x^4+(6a+6)x^3+(3a+2)x^2+(6a+6)x+5a+4$
- $y^2=(4a+4)x^6+(2a+3)x^5+6x^4+(3a+1)x^3+(5a+5)x^2+6ax+6a$
- $y^2=5x^6+(4a+3)x^5+ax^4+3x^3+(3a+2)x^2+(a+1)x+3a+5$
- $y^2=2x^6+(3a+3)x^5+(5a+3)x^4+(a+3)x^3+(5a+1)x^2+(4a+4)x+a+1$
- $y^2=6ax^6+(5a+5)x^5+(5a+2)x^4+(2a+4)x^3+(3a+6)x^2+(4a+2)x+2a+3$
- $y^2=4ax^6+(4a+6)x^5+(2a+6)x^4+(6a+2)x^3+(a+2)x^2+6ax+3a+2$
- $y^2=4ax^6+(4a+3)x^5+4x^4+(3a+2)x^3+(6a+2)x^2+(a+4)x+4a+3$
- $y^2=(4a+3)x^6+(a+1)x^5+(2a+5)x^4+(5a+6)x^3+x^2+(a+2)x+2a+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$The isogeny class factors as 1.49.am $\times$ 1.49.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ad_ak | $2$ | (not in LMFDB) |
2.49.d_ak | $2$ | (not in LMFDB) |
2.49.v_hy | $2$ | (not in LMFDB) |