Invariants
| Base field: | $\F_{137}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 22 x + 137 x^{2} )( 1 - 20 x + 137 x^{2} )$ |
| $1 - 42 x + 714 x^{2} - 5754 x^{3} + 18769 x^{4}$ | |
| Frobenius angles: | $\pm0.111017258455$, $\pm0.173950867407$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $13688$ | $346032640$ | $6608297143352$ | $124103249888051200$ | $2329217153377091196728$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $96$ | $18434$ | $2569968$ | $352290462$ | $48262203216$ | $6611863908386$ | $905824399026048$ | $124097930864479678$ | $17001416412132284736$ | $2329194047588225099714$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=48x^6+28x^5+64x^4+3x^3+7x^2+39x+33$
- $y^2=91x^6+23x^5+89x^4+30x^3+89x^2+23x+91$
- $y^2=136x^6+20x^5+51x^4+30x^3+51x^2+20x+136$
- $y^2=92x^6+81x^5+2x^4+31x^3+2x^2+81x+92$
- $y^2=30x^6+69x^5+57x^4+127x^3+79x^2+129x+63$
- $y^2=52x^6+82x^5+31x^4+44x^3+114x^2+43x+66$
- $y^2=131x^6+119x^5+62x^4+3x^3+62x^2+119x+131$
- $y^2=28x^6+68x^5+106x^4+56x^3+54x^2+93x+30$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{137}$.
Endomorphism algebra over $\F_{137}$| The isogeny class factors as 1.137.aw $\times$ 1.137.au 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.