Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 29 x^{2} )( 1 - 6 x + 29 x^{2} )$ |
$1 - 15 x + 112 x^{2} - 435 x^{3} + 841 x^{4}$ | |
Frobenius angles: | $\pm0.185103371333$, $\pm0.311919362152$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $12$ |
Isomorphism classes: | 60 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $504$ | $707616$ | $603669024$ | $501912028800$ | $420853401410904$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $15$ | $841$ | $24750$ | $709633$ | $20518275$ | $594824326$ | $17249813295$ | $500246477473$ | $14507148445830$ | $420707234868481$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=15x^6+10x^5+8x^4+23x^3+28x^2+8x+16$
- $y^2=12x^6+23x^5+x^4+10x^3+22x^2+2x+21$
- $y^2=15x^6+2x^5+9x^4+x^3+27x^2+9x+27$
- $y^2=21x^6+6x^5+16x^4+3x^3+6x^2+5x+2$
- $y^2=11x^6+22x^5+2x^4+3x^3+x^2+13x+24$
- $y^2=18x^6+18x^5+19x^4+6x^3+7x^2+9x+18$
- $y^2=13x^6+20x^5+4x^4+17x^3+18x^2+7x+26$
- $y^2=19x^6+27x^5+25x^4+17x^3+15x^2+3x+25$
- $y^2=5x^6+3x^5+10x^4+4x^3+28x^2+13x+18$
- $y^2=10x^6+4x^5+20x^4+28x^3+21x^2+5x+18$
- $y^2=17x^6+12x^5+4x^4+18x^3+21x^2+8x+18$
- $y^2=10x^6+10x^5+7x^4+5x^3+4x^2+9x+24$
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.ag 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.ad_e | $2$ | (not in LMFDB) |
2.29.d_e | $2$ | (not in LMFDB) |
2.29.p_ei | $2$ | (not in LMFDB) |