Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 29 x^{2} )( 1 - 7 x + 29 x^{2} )$ |
$1 - 16 x + 121 x^{2} - 464 x^{3} + 841 x^{4}$ | |
Frobenius angles: | $\pm0.185103371333$, $\pm0.274796655058$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $483$ | $696969$ | $602691264$ | $502197528105$ | $420959674288923$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $14$ | $828$ | $24710$ | $710036$ | $20523454$ | $594847206$ | $17249778406$ | $500245349476$ | $14507141540990$ | $420707225510028$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=28x^6+22x^5+18x^4+6x^3+18x^2+22x+28$
- $y^2=22x^6+19x^5+5x^4+2x^3+5x^2+19x+22$
- $y^2=26x^6+10x^5+9x^4+5x^3+9x^2+10x+26$
- $y^2=3x^6+10x^5+9x^4+26x^3+9x^2+10x+3$
- $y^2=15x^6+12x^5+6x^4+13x^3+6x^2+12x+15$
- $y^2=12x^6+26x^5+2x^4+28x^3+2x^2+26x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$The isogeny class factors as 1.29.aj $\times$ 1.29.ah 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 |
---|---|---|
2.29.ac_af | $2$ | (not in LMFDB) |
2.29.c_af | $2$ | (not in LMFDB) |
2.29.q_er | $2$ | (not in LMFDB) |