Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 14 x + 97 x^{2} - 406 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.107469610167$, $\pm0.384030083410$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2062400.2 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $519$ | $705321$ | $597557916$ | $499853234169$ | $420549042007599$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $840$ | $24502$ | $706724$ | $20503436$ | $594816606$ | $17250209804$ | $500249070724$ | $14507154596398$ | $420707237231400$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=19x^6+x^4+20x^3+2x^2+6x+23$
- $y^2=12x^6+24x^5+9x^4+27x^3+3x^2+5x+22$
- $y^2=8x^6+26x^5+14x^4+10x^3+19x^2+18x+3$
- $y^2=8x^6+24x^5+23x^4+9x^3+12x^2+8x+21$
- $y^2=13x^6+25x^5+26x^4+22x^3+24x^2+26x+21$
- $y^2=14x^6+6x^5+10x^4+15x^3+13x^2+16x+8$
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 4.0.2062400.2. |
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.o_dt | $2$ | (not in LMFDB) |