Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 43 x + 737 x^{2} - 5977 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0488276070962$, $\pm0.185265935553$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.368589.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14039$ | $366123081$ | $7206201614537$ | $139351783069244061$ | $2692458192750605660624$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $97$ | $18947$ | $2683255$ | $373295995$ | $51888960112$ | $7212551205287$ | $1002544375960357$ | $139353667002050179$ | $19370159736341116855$ | $2692452204100261958102$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=74x^6+16x^5+42x^4+133x^3+86x^2+100x+40$
- $y^2=138x^6+136x^5+81x^4+78x^3+94x^2+33x+19$
- $y^2=93x^6+97x^5+35x^4+34x^3+70x^2+54x+126$
- $y^2=18x^6+77x^5+101x^4+x^3+29x^2+87x+32$
- $y^2=81x^6+77x^5+121x^4+51x^3+7x^2+4x+115$
- $y^2=61x^6+124x^5+38x^4+73x^3+102x^2+96x+19$
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 4.0.368589.1. |
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_bcj | $2$ | (not in LMFDB) |