Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 14 x + 104 x^{2} - 406 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.199059286213$, $\pm0.337319176989$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.505152.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $526$ | $718516$ | $604803742$ | $501745470928$ | $420790217052766$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $854$ | $24796$ | $709398$ | $20515196$ | $594814646$ | $17249851712$ | $500246834014$ | $14507147503648$ | $420707208209414$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=21x^6+12x^5+18x^4+25x^3+18x^2+4x+2$
- $y^2=3x^6+24x^5+13x^4+23x^3+14x^2+12x+8$
- $y^2=2x^6+6x^5+18x^4+12x^3+4x^2+8x+14$
- $y^2=20x^6+10x^5+23x^4+7x^3+6x^2+25x+27$
- $y^2=24x^6+27x^5+6x^3+12x^2+x+15$
- $y^2=17x^6+9x^5+11x^4+28x^3+6x^2+27x+15$
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 4.0.505152.1. |
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_ea | $2$ | (not in LMFDB) |