Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 86 x^{2} - 377 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.0969199097066$, $\pm0.418576947707$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3677868.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $538$ | $709084$ | $595464856$ | $499217826688$ | $420520360012738$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $17$ | $845$ | $24416$ | $705825$ | $20502037$ | $594837434$ | $17250272329$ | $500248137313$ | $14507147319056$ | $420707239898405$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=27x^6+26x^5+16x^4+28x^3+11x^2+9x+11$
- $y^2=14x^5+16x^4+3x^3+25x^2+26x+27$
- $y^2=26x^6+18x^5+27x^4+11x^3+6x^2+11x$
- $y^2=2x^6+8x^5+14x^4+25x^3+28x^2+3$
- $y^2=13x^5+27x^4+8x^3+4x^2+21x+2$
- $y^2=3x^6+13x^5+7x^4+25x^3+5x^2+23x+11$
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 4.0.3677868.2. |
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.n_di | $2$ | (not in LMFDB) |