Invariants
Base field: | $\F_{137}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 39 x + 644 x^{2} - 5343 x^{3} + 18769 x^{4}$ |
Frobenius angles: | $\pm0.0784760750967$, $\pm0.254857258237$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{41})\) |
Galois group: | $C_2^2$ |
Jacobians: | $32$ |
This isogeny class is simple but not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14032$ | $347937472$ | $6611855780416$ | $124103830860589824$ | $2329200821682127388752$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $99$ | $18537$ | $2571354$ | $352292113$ | $48261864819$ | $6611855310222$ | $905824268766411$ | $124097929543725601$ | $17001416405572203978$ | $2329194047653348323657$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=35x^6+134x^5+47x^4+14x^3+109x^2+85x+19$
- $y^2=100x^6+95x^5+134x^4+116x^3+52x^2+36x+80$
- $y^2=47x^6+119x^5+20x^4+136x^3+72x^2+74x+111$
- $y^2=94x^6+120x^5+14x^4+85x^3+56x^2+120x+62$
- $y^2=95x^6+11x^5+x^4+124x^3+117x^2+10x+58$
- $y^2=110x^6+63x^5+66x^4+77x^3+26x^2+14x+3$
- $y^2=57x^6+46x^5+104x^4+22x^3+6x^2+40x+104$
- $y^2=32x^6+14x^5+88x^4+19x^3+13x^2+49x+83$
- $y^2=22x^6+116x^5+88x^4+30x^3+31x^2+47x+108$
- $y^2=131x^6+38x^5+113x^4+116x^3+122x^2+78x+92$
- $y^2=104x^6+x^5+103x^4+17x^3+62x^2+105x+25$
- $y^2=24x^6+81x^5+108x^4+40x^3+132x^2+79x+110$
- $y^2=13x^6+81x^5+100x^4+97x^3+60x^2+12x+51$
- $y^2=85x^6+107x^5+24x^4+89x^3+3x^2+12x+62$
- $y^2=67x^6+95x^5+29x^3+73x^2+134x+127$
- $y^2=73x^6+88x^5+98x^4+111x^3+136x^2+45x+116$
- $y^2=39x^6+29x^5+107x^4+119x^3+89x^2+72x+94$
- $y^2=84x^6+51x^5+29x^3+90x^2+73x+128$
- $y^2=100x^6+56x^5+119x^4+120x^3+86x^2+78x+131$
- $y^2=23x^6+76x^5+37x^4+61x^3+2x^2+62x+23$
- and 12 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{137^{6}}$.
Endomorphism algebra over $\F_{137}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{41})\). |
The base change of $A$ to $\F_{137^{6}}$ is 1.6611856250609.abatok 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$ |
- Endomorphism algebra over $\F_{137^{2}}$
The base change of $A$ to $\F_{137^{2}}$ is the simple isogeny class 2.18769.aiz_caoe and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\). - Endomorphism algebra over $\F_{137^{3}}$
The base change of $A$ to $\F_{137^{3}}$ is the simple isogeny class 2.2571353.a_abatok and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\).
Base change
This is a primitive isogeny class.