Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 42 x + 716 x^{2} - 5838 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0856055396078$, $\pm0.195556570621$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1044288.8 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14158$ | $366947044$ | $7208830516486$ | $139357993221025488$ | $2692470486666064292038$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $98$ | $18990$ | $2684234$ | $373312630$ | $51889197038$ | $7212554237454$ | $1002544413205286$ | $139353667471136158$ | $19370159742618181154$ | $2692452204187518300030$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=74x^6+103x^5+25x^4+21x^3+77x^2+69x+23$
- $y^2=64x^6+29x^4+121x^3+48x^2+131x+104$
- $y^2=26x^6+115x^5+53x^4+101x^3+58x^2+34x$
- $y^2=125x^6+71x^5+40x^4+28x^3+89x^2+57x+70$
- $y^2=102x^6+59x^5+112x^4+21x^3+39x^2+125x+78$
- $y^2=51x^6+129x^5+76x^4+99x^3+71x^2+75x+31$
- $y^2=23x^6+2x^5+22x^4+x^3+40x^2+123x+94$
- $y^2=91x^6+15x^5+105x^4+28x^3+106x^2+85x+63$
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 4.0.1044288.8. |
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.bq_bbo | $2$ | (not in LMFDB) |