Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 15 x + 113 x^{2} - 435 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.204745548845$, $\pm0.298120994119$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.78525.3 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $505$ | $709525$ | $604782445$ | $502230885525$ | $420898030432000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $15$ | $843$ | $24795$ | $710083$ | $20520450$ | $594821643$ | $17249678055$ | $500245318723$ | $14507143413435$ | $420707233683798$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+5x^5+5x^4+3x^3+17x^2+24x+10$
- $y^2=23x^6+25x^5+26x^4+10x^3+22x^2+19x+15$
- $y^2=2x^6+7x^5+26x^4+17x^3+24x^2+9x+10$
- $y^2=18x^6+5x^5+14x^4+27x^3+24x^2+9x+27$
- $y^2=13x^6+19x^5+13x^4+26x^3+9x^2+4x+17$
- $y^2=8x^6+9x^5+22x^4+6x^3+18x^2+23$
- $y^2=19x^6+5x^5+12x^4+9x^3+22x^2+4x+18$
- $y^2=28x^6+x^5+16x^4+16x^3+16x^2+13x+14$
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 4.0.78525.3. |
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.p_ej | $2$ | (not in LMFDB) |