Invariants
Base field: | $\F_{137}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 41 x + 690 x^{2} - 5617 x^{3} + 18769 x^{4}$ |
Frobenius angles: | $\pm0.0859306910897$, $\pm0.211296820890$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2343212.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13802$ | $346678636$ | $6609539135816$ | $124103385773447936$ | $2329210271231295013402$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $97$ | $18469$ | $2570452$ | $352290849$ | $48262060617$ | $6611859930034$ | $905824329160225$ | $124097929962305793$ | $17001416403959318836$ | $2329194047560131055029$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=105x^6+117x^5+87x^4+65x^3+135x^2+58$
- $y^2=122x^6+130x^5+113x^4+30x^3+2x^2+40x+89$
- $y^2=67x^6+33x^5+43x^4+63x^3+99x^2+68x+49$
- $y^2=79x^6+58x^5+120x^4+26x^3+4x^2+136x+99$
- $y^2=67x^6+56x^5+65x^4+4x^3+47x^2+103x+116$
- $y^2=x^6+41x^5+37x^4+115x^3+69x^2+43x+64$
- $y^2=115x^6+83x^5+126x^4+52x^3+81x^2+37x+75$
- $y^2=121x^6+125x^5+28x^4+93x^3+133x^2+5x+6$
- $y^2=79x^6+84x^5+63x^4+51x^3+12x^2+12x+131$
- $y^2=86x^6+34x^5+116x^4+37x^3+99x^2+132x+114$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{137}$.
Endomorphism algebra over $\F_{137}$The endomorphism algebra of this simple isogeny class is 4.0.2343212.1. |
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 |
---|---|---|
2.137.bp_bao | $2$ | (not in LMFDB) |