Properties

 Label 2.64.abe_np Base Field $\F_{2^{6}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $( 1 - 15 x + 64 x^{2} )^{2}$ Frobenius angles: $\pm0.113134082257$, $\pm0.113134082257$ Angle rank: $1$ (numerical) Jacobians: 3

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

• $y^2+(x^2+x)y=(a^5+a)x^5+(a^3+a^2)x^4+(a^5+a)x^3+(a^4+a^3+a+1)x^2+(a^4+a^2+a+1)x$
• $y^2+(x^2+x)y=(a^5+a^4+a^2+1)x^5+(a^5+a^3+a^2)x^4+(a^5+a^4+a^2+1)x^3+(a^3+a^2+a)x^2+(a^5+a)x$
• $y^2+(x^2+x)y=(a^4+a^2+a+1)x^5+(a^3+a+1)x^4+(a^4+a^2+a+1)x^3+(a^5+a^4+a^3+a^2+a)x^2+(a^5+a^4+a^2+1)x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2500 16000000 68460722500 281434176000000 1152950336902562500 4722404864699536000000 19342842402600710328722500 79228180569952877158656000000 324518563314017073918940214402500 1329228000321092509518790810000000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 35 3903 261155 16774783 1073768675 68720035263 4398053170595 281475040857343 18014399045474915 1152921508541353023

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The isogeny class factors as 1.64.ap 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-31})$$$)$
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.a_adt $2$ (not in LMFDB) 2.64.be_np $2$ (not in LMFDB) 2.64.p_gf $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.a_adt $2$ (not in LMFDB) 2.64.be_np $2$ (not in LMFDB) 2.64.p_gf $3$ (not in LMFDB) 2.64.a_dt $4$ (not in LMFDB) 2.64.ap_gf $6$ (not in LMFDB)