Invariants
Base field: | $\F_{29}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 15 x + 111 x^{2} - 435 x^{3} + 841 x^{4}$ |
Frobenius angles: | $\pm0.168115237638$, $\pm0.322576897349$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.415909.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $503$ | $705709$ | $602556275$ | $501590434549$ | $420805702851728$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $15$ | $839$ | $24705$ | $709179$ | $20515950$ | $594824303$ | $17249918505$ | $500247429619$ | $14507152932015$ | $420707242323254$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=21x^6+16x^5+27x^4+25x^3+6x^2+21x+12$
- $y^2=3x^6+15x^5+20x^4+21x^3+22x^2+26x+26$
- $y^2=5x^6+7x^5+8x^4+5x^3+3x^2+23x+7$
- $y^2=9x^6+7x^5+17x^4+22x^3+26x^2+14$
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 4.0.415909.1. |
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.p_eh | $2$ | (not in LMFDB) |