Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 14 x + 105 x^{2} - 406 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.214587444740$, $\pm0.326443249102$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.245312.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $527$ | $720409$ | $605841308$ | $502004764697$ | $420813202681447$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $856$ | $24838$ | $709764$ | $20516316$ | $594805150$ | $17249710396$ | $500246008260$ | $14507146008574$ | $420707224173336$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+5x^5+27x^4+19x^3+28x^2+23x+11$
- $y^2=18x^6+10x^5+10x^4+20x^3+17x^2+21x+2$
- $y^2=5x^6+2x^5+26x^4+3x^3+6x+20$
- $y^2=25x^6+17x^5+14x^4+x^3+12x^2+x+3$
- $y^2=11x^6+8x^5+22x^4+2x^3+3x^2+2x+27$
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 4.0.245312.1. |
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_eb | $2$ | (not in LMFDB) |