Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 15 x + 61 x^{2} )( 1 - 11 x + 61 x^{2} )$ |
$1 - 26 x + 287 x^{2} - 1586 x^{3} + 3721 x^{4}$ | |
Frobenius angles: | $\pm0.0900194921159$, $\pm0.251304563322$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $12$ |
Isomorphism classes: | 20 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2397$ | $13473537$ | $51532201728$ | $191766494252025$ | $713369242431934077$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $36$ | $3620$ | $227034$ | $13850116$ | $844627476$ | $51520420262$ | $3142741667556$ | $191707302982276$ | $11694146124592674$ | $713342913338930180$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=27x^6+2x^5+52x^4+53x^3+3x^2+7x+60$
- $y^2=x^6+24x^5+53x^4+22x^3+25x^2+49x+24$
- $y^2=45x^6+22x^5+23x^4+23x^3+11x^2+56x+15$
- $y^2=35x^6+23x^5+39x^4+22x^3+13x^2+54x+21$
- $y^2=38x^6+40x^5+12x^4+23x^3+49x^2+40x+23$
- $y^2=5x^6+54x^5+31x^4+29x^3+30x+24$
- $y^2=51x^6+29x^5+19x^4+23x^3+46x^2+22x+50$
- $y^2=32x^6+42x^5+39x^4+38x^3+45x^2+57x+30$
- $y^2=44x^6+45x^5+35x^4+17x^3+58x^2+6x+57$
- $y^2=40x^6+13x^5+11x^4+54x^3+45x^2+33x+54$
- $y^2=24x^6+41x^5+52x^4+35x^3+58x^2+52x+28$
- $y^2=53x^6+33x^5+43x^4+13x^3+55x^2+24x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$The isogeny class factors as 1.61.ap $\times$ 1.61.al 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 |
---|---|---|
2.61.ae_abr | $2$ | (not in LMFDB) |
2.61.e_abr | $2$ | (not in LMFDB) |
2.61.ba_lb | $2$ | (not in LMFDB) |