Invariants
Base field: | $\F_{137}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 40 x + 666 x^{2} - 5480 x^{3} + 18769 x^{4}$ |
Frobenius angles: | $\pm0.0710640823629$, $\pm0.237869879993$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.27200.2 |
Galois group: | $D_{4}$ |
Jacobians: | $44$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13916$ | $347287696$ | $6610519275164$ | $124102708061889536$ | $2329202300344601817436$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $98$ | $18502$ | $2570834$ | $352288926$ | $48261895458$ | $6611856237478$ | $905824276012754$ | $124097929446096318$ | $17001416401620954338$ | $2329194047586806054022$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 44 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=54x^6+49x^5+76x^4+86x^3+87x^2+112x+60$
- $y^2=102x^6+18x^5+134x^4+20x^3+43x^2+118x+43$
- $y^2=110x^6+128x^5+39x^4+41x^3+37x^2+84x+134$
- $y^2=118x^6+95x^5+27x^4+60x^3+125x^2+46x+128$
- $y^2=20x^6+103x^5+101x^4+86x^3+35x^2+60x+26$
- $y^2=126x^6+50x^5+67x^4+55x^3+127x^2+99x+17$
- $y^2=130x^6+93x^5+130x^4+131x^3+111x^2+10x+5$
- $y^2=83x^6+33x^5+118x^4+135x^3+x^2+96x+133$
- $y^2=42x^6+48x^5+5x^4+98x^3+99x^2+59x+106$
- $y^2=17x^6+62x^5+118x^4+63x^3+80x^2+54x+118$
- $y^2=51x^6+113x^5+43x^4+20x^3+9x^2+7x+120$
- $y^2=49x^6+85x^5+35x^4+44x^3+132x^2+127x+66$
- $y^2=126x^6+71x^5+48x^4+64x^3+12x^2+111x+134$
- $y^2=54x^6+63x^5+45x^4+35x^3+109x^2+112x+27$
- $y^2=82x^6+37x^5+20x^4+74x^3+27x^2+x+27$
- $y^2=132x^6+102x^5+39x^4+14x^3+36x^2+54x+57$
- $y^2=44x^6+70x^5+66x^4+60x^3+129x^2+53x+79$
- $y^2=26x^6+50x^5+111x^4+22x^3+105x^2+114x+77$
- $y^2=119x^6+13x^5+89x^4+38x^3+15x^2+51x+80$
- $y^2=117x^6+58x^5+74x^4+98x^3+87x^2+23x+87$
- and 24 more
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 4.0.27200.2. |
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.bo_zq | $2$ | (not in LMFDB) |