Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 44 x + 760 x^{2} - 6116 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0377284713884$, $\pm0.162150612510$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.65792.3 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13922$ | $365341124$ | $7203929001458$ | $139347325722259472$ | $2692452179043897658722$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $96$ | $18906$ | $2682408$ | $373284054$ | $51888844216$ | $7212550695690$ | $1002544383388896$ | $139353667220446110$ | $19370159739293638944$ | $2692452204119256352186$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=115x^6+117x^5+115x^4+51x^3+17x^2+51x+33$
- $y^2=137x^6+9x^5+117x^4+126x^3+103x^2+77x+8$
- $y^2=7x^6+133x^5+72x^4+85x^3+54x^2+107x+72$
- $y^2=15x^6+118x^5+39x^4+119x^3+61x^2+8x+43$
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 4.0.65792.3. |
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.bs_bdg | $2$ | (not in LMFDB) |