# Properties

 Label 2.64.az_km Base Field $\F_{2^{6}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $( 1 - 8 x )^{2}( 1 - 9 x + 64 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.309839631512$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2744 16447536 68712424424 281437900380000 1152840305690007704 4722297901148943205776 19342778756208740794944584 79228152438505363164788280000 324518552630200417591949193187064 1329227995667562387786429070824101616

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 40 4016 262120 16775008 1073666200 68718478736 4398038699080 281474940914368 18014398452403960 1152921504505059056

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The isogeny class factors as 1.64.aq $\times$ 1.64.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.64.aq : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.64.aj : $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.ah_aq $2$ (not in LMFDB) 2.64.h_aq $2$ (not in LMFDB) 2.64.z_km $2$ (not in LMFDB) 2.64.ab_ce $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.ah_aq $2$ (not in LMFDB) 2.64.h_aq $2$ (not in LMFDB) 2.64.z_km $2$ (not in LMFDB) 2.64.ab_ce $3$ (not in LMFDB) 2.64.aj_ey $4$ (not in LMFDB) 2.64.j_ey $4$ (not in LMFDB) 2.64.ar_hs $6$ (not in LMFDB) 2.64.b_ce $6$ (not in LMFDB) 2.64.r_hs $6$ (not in LMFDB)