Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 17 x + 127 x^{2} - 493 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.0941348138985$, $\pm0.286392861421$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.118469.1 |
Galois group: | $D_{4}$ |
Jacobians: | $5$ |
Isomorphism classes: | 5 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $459$ | $678861$ | $596889567$ | $500864324661$ | $420729588852624$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $807$ | $24475$ | $708155$ | $20512238$ | $594802287$ | $17249739443$ | $500246559859$ | $14507154569965$ | $420707310958902$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=25x^6+3x^5+19x^4+20x^3+23x^2+12x+26$
- $y^2=18x^6+25x^5+20x^4+7x^3+28x^2+10x+17$
- $y^2=27x^6+24x^5+7x^4+20x^3+13x^2+2x+8$
- $y^2=12x^6+8x^5+24x^4+23x^3+5x^2+3x+21$
- $y^2=18x^6+15x^5+2x^4+16x^3+11x^2+5x+20$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$The endomorphism algebra of this simple isogeny class is 4.0.118469.1. |
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 |
---|---|---|
2.29.r_ex | $2$ | (not in LMFDB) |