Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 16 x + 71 x^{2} )( 1 - 13 x + 71 x^{2} )$ |
$1 - 29 x + 350 x^{2} - 2059 x^{3} + 5041 x^{4}$ | |
Frobenius angles: | $\pm0.101666819831$, $\pm0.219552767034$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3304$ | $24713920$ | $128059088416$ | $645917180634880$ | $3255393668484748504$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $43$ | $4901$ | $357796$ | $25418121$ | $1804312553$ | $128100915038$ | $9095123183063$ | $645753536922481$ | $45848500704490636$ | $3255243551664577901$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=69x^6+10x^5+62x^4+60x^3+57x^2+10x+55$
- $y^2=51x^6+65x^5+46x^4+11x^3+69x^2+38x+7$
- $y^2=7x^6+26x^5+22x^4+15x^3+29x^2+23x+34$
- $y^2=34x^6+44x^5+24x^4+26x^3+57x^2+68x+57$
- $y^2=33x^6+32x^5+2x^4+57x^3+66x^2+15x+66$
- $y^2=17x^6+68x^5+69x^4+8x^3+19x^2+20x$
- $y^2=65x^6+20x^5+2x^4+34x^3+36x^2+50x+48$
- $y^2=14x^6+67x^5+2x^4+33x^3+34x^2+13x+18$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71}$.
Endomorphism algebra over $\F_{71}$The isogeny class factors as 1.71.aq $\times$ 1.71.an 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 |
---|---|---|
2.71.ad_aco | $2$ | (not in LMFDB) |
2.71.d_aco | $2$ | (not in LMFDB) |
2.71.bd_nm | $2$ | (not in LMFDB) |