Invariants
Base field: | $\F_{137}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 41 x + 689 x^{2} - 5617 x^{3} + 18769 x^{4}$ |
Frobenius angles: | $\pm0.0733101034139$, $\pm0.216316581678$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2725821.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13801$ | $346639717$ | $6609222180241$ | $124101988459622629$ | $2329205927883038252176$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $97$ | $18467$ | $2570329$ | $352286883$ | $48261970622$ | $6611858334947$ | $905824306135937$ | $124097929689940291$ | $17001416401408406953$ | $2329194047543808886382$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+71x^5+68x^4+123x^3+7x^2+73x+54$
- $y^2=36x^6+17x^5+115x^4+105x^3+116x^2+59x+49$
- $y^2=12x^6+76x^5+10x^4+109x^3+85x^2+63x+74$
- $y^2=116x^6+29x^5+134x^4+129x^3+48x^2+125x+43$
- $y^2=82x^6+115x^5+5x^4+62x^3+97x+55$
- $y^2=97x^6+26x^5+83x^4+61x^3+55x^2+4x+115$
- $y^2=111x^6+53x^5+96x^4+135x^3+53x^2+127x+62$
- $y^2=108x^6+16x^5+112x^4+38x^3+29x^2+36x+84$
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 4.0.2725821.1. |
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_ban | $2$ | (not in LMFDB) |