Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 43 x + 739 x^{2} - 5977 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0912189096588$, $\pm0.167706197790$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.156125.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14041$ | $366203321$ | $7206896114371$ | $139355101493153645$ | $2692469616719167387696$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $97$ | $18951$ | $2683513$ | $373304883$ | $51889180272$ | $7212555578811$ | $1002544449231583$ | $139353668061265683$ | $19370159749611727147$ | $2692452204241998470806$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=133x^6+102x^5+81x^4+13x^3+54x^2+49x+22$
- $y^2=10x^6+62x^5+113x^4+104x^3+69x^2+22x+132$
- $y^2=81x^6+93x^5+126x^4+27x^3+75x^2+102x+15$
- $y^2=56x^6+68x^5+26x^4+57x^3+99x^2+109x+85$
- $y^2=125x^6+123x^5+105x^4+87x^3+84x^2+65x+43$
- $y^2=19x^6+30x^5+17x^4+127x^3+11x^2+110x+73$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{139}$.
Endomorphism algebra over $\F_{139}$The endomorphism algebra of this simple isogeny class is 4.0.156125.1. |
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 |
---|---|---|
2.139.br_bcl | $2$ | (not in LMFDB) |