Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 14 x + 94 x^{2} - 406 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.0557514217637$, $\pm0.397939584956$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.62192.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $516$ | $699696$ | $594461412$ | $499000796928$ | $420402654216516$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $834$ | $24376$ | $705518$ | $20496296$ | $594782226$ | $17249992832$ | $500247360094$ | $14507144089888$ | $420707199735714$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+11x^5+21x^3+11x^2+17x+12$
- $y^2=22x^6+x^5+21x^4+9x^3+21x^2+23x+2$
- $y^2=18x^6+15x^5+18x^4+4x^3+11x^2+24x+15$
- $y^2=18x^6+27x^4+14x^3+20x^2+24x+14$
- $y^2=28x^6+26x^5+5x^4+22x^3+8x^2+15x+14$
- $y^2=3x^6+21x^5+10x^3+19x^2+27x+16$
- $y^2=27x^6+4x^5+20x^4+3x^3+8x^2+28x+15$
- $y^2=19x^6+5x^5+13x^4+28x^3+10x^2+25x+13$
- $y^2=5x^6+5x^5+3x^4+3x^3+7x^2+28x$
- $y^2=16x^6+11x^5+2x^4+10x^3+4x^2+16x+11$
- $y^2=19x^6+15x^5+14x^4+24x^3+22x^2+21x+18$
- $y^2=3x^6+28x^5+24x^4+14x^3+18x^2+20x+9$
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 4.0.62192.1. |
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.o_dq | $2$ | (not in LMFDB) |