Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 29 x^{2} )( 1 - 5 x + 29 x^{2} )$ |
$1 - 14 x + 103 x^{2} - 406 x^{3} + 841 x^{4}$ | |
Frobenius angles: | $\pm0.185103371333$, $\pm0.346328109963$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $24$ |
Isomorphism classes: | 40 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $525$ | $716625$ | $603766800$ | $501483425625$ | $420764365300125$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $852$ | $24754$ | $709028$ | $20513936$ | $594821862$ | $17249971664$ | $500247561028$ | $14507149291546$ | $420707201410452$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=20x^6+25x^5+15x^4+9x^3+3x^2+x+28$
- $y^2=20x^6+14x^5+19x^3+14x+9$
- $y^2=15x^6+28x^5+13x^4+5x^3+8x^2+4x+3$
- $y^2=18x^6+3x^5+27x^4+11x^3+15x^2+17x+11$
- $y^2=14x^6+10x^5+22x^4+20x^3+7x^2+10x+15$
- $y^2=2x^6+17x^5+26x^4+20x^3+14x^2+26x+19$
- $y^2=20x^6+8x^5+13x^4+25x^3+24x^2+5x+27$
- $y^2=23x^6+4x^5+4x^4+14x^3+10x^2+25x+4$
- $y^2=17x^6+11x^5+17x^4+24x^3+4x^2+19x+28$
- $y^2=6x^6+12x^5+27x^4+25x^3+7x^2+15$
- $y^2=15x^6+7x^5+15x^4+3x^3+18x^2+24x+12$
- $y^2=12x^6+13x^5+13x^4+8x^3+11x^2+18x+28$
- $y^2=6x^6+5x^5+24x^4+28x^3+28x^2+25x+3$
- $y^2=16x^6+20x^5+14x^4+24x^3+11x^2+23x+7$
- $y^2=3x^6+6x^5+14x^4+5x^3+12x^2+5x+27$
- $y^2=14x^6+2x^5+8x^4+20x^3+16x^2+17x+27$
- $y^2=11x^6+9x^5+2x^4+13x^3+2x^2+9x+11$
- $y^2=19x^6+24x^5+12x^4+27x^3+17x^2+24x+10$
- $y^2=18x^6+20x^5+21x^4+13x^3+18x^2+7x+17$
- $y^2=5x^6+8x^5+11x^4+24x^3+19x^2+15x+4$
- $y^2=17x^6+14x^5+27x^4+9x^3+x^2+18x+19$
- $y^2=5x^6+21x^5+5x^3+14x+25$
- $y^2=14x^6+19x^5+13x^4+17x^3+26x^2+22x+17$
- $y^2=3x^6+4x^5+17x^4+17x^3+21x^2+5x+17$
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.af 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.ae_n | $2$ | (not in LMFDB) |
2.29.e_n | $2$ | (not in LMFDB) |
2.29.o_dz | $2$ | (not in LMFDB) |