Invariants
Base field: | $\F_{137}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 23 x + 137 x^{2} )( 1 - 17 x + 137 x^{2} )$ |
$1 - 40 x + 665 x^{2} - 5480 x^{3} + 18769 x^{4}$ | |
Frobenius angles: | $\pm0.0596181899068$, $\pm0.241282780251$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $44$ |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13915$ | $347248825$ | $6610210097920$ | $124101391104315625$ | $2329198381471590022075$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $98$ | $18500$ | $2570714$ | $352285188$ | $48261814258$ | $6611854868750$ | $905824257200674$ | $124097929228986628$ | $17001416399439692618$ | $2329194047566011402500$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 44 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+67x^5+66x^4+129x^3+66x^2+67x+14$
- $y^2=4x^6+64x^5+99x^4+74x^3+99x^2+64x+4$
- $y^2=120x^6+82x^5+18x^4+130x^3+94x^2+74x+42$
- $y^2=13x^6+10x^5+107x^4+58x^3+99x^2+9x+72$
- $y^2=93x^6+27x^5+6x^4+25x^3+6x^2+27x+93$
- $y^2=111x^6+34x^5+44x^4+81x^3+36x^2+79x+47$
- $y^2=41x^6+34x^5+36x^4+134x^3+50x^2+2x+88$
- $y^2=103x^6+118x^5+28x^4+19x^3+28x^2+118x+103$
- $y^2=128x^6+121x^5+37x^4+53x^3+60x^2+74x+98$
- $y^2=47x^6+12x^5+58x^4+119x^3+123x^2+39x+122$
- $y^2=42x^6+4x^5+55x^4+23x^3+17x^2+121x+37$
- $y^2=41x^6+41x^4+37x^3+69x^2+78x+83$
- $y^2=89x^6+129x^5+75x^4+102x^3+5x^2+10x+126$
- $y^2=125x^6+125x^5+86x^4+67x^3+68x^2+18x+94$
- $y^2=130x^6+24x^5+33x^4+42x^3+33x^2+24x+130$
- $y^2=75x^6+67x^5+91x^4+18x^3+91x^2+67x+75$
- $y^2=71x^6+97x^5+85x^4+130x^3+85x^2+97x+71$
- $y^2=51x^6+111x^5+136x^4+70x^3+35x^2+23x+79$
- $y^2=9x^6+59x^5+99x^4+127x^3+41x^2+19x+23$
- $y^2=61x^6+24x^5+71x^4+24x^3+120x^2+65x+89$
- and 24 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{137}$.
Endomorphism algebra over $\F_{137}$The isogeny class factors as 1.137.ax $\times$ 1.137.ar and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ag_aen | $2$ | (not in LMFDB) |
2.137.g_aen | $2$ | (not in LMFDB) |
2.137.bo_zp | $2$ | (not in LMFDB) |