# Properties

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

## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $( 1 - 15 x + 64 x^{2} )( 1 - 8 x + 64 x^{2} )$ Frobenius angles: $\pm0.113134082257$, $\pm0.333333333333$ Angle rank: $1$ (numerical) Jacobians: 6

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2850 16644000 68858168850 281523306888000 1152900736926299250 4722349644870882444000 19342818534837223169380050 79228176264475395472476432000 324518563321925232852664256646450 1329227999290944231826880336654100000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 42 4064 262674 16780096 1073722482 68719231712 4398047743698 281475025561216 18014399045913906 1152921507647841824

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The isogeny class factors as 1.64.ap $\times$ 1.64.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{6}}$
 The base change of $A$ to $\F_{2^{18}}$ is 1.262144.atb $\times$ 1.262144.bnk. The endomorphism algebra for each factor is: 1.262144.atb : $$\Q(\sqrt{-31})$$. 1.262144.bnk : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{18}}$.

## 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_i $2$ (not in LMFDB) 2.64.h_i $2$ (not in LMFDB) 2.64.x_jo $2$ (not in LMFDB) 2.64.b_aei $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.ah_i $2$ (not in LMFDB) 2.64.h_i $2$ (not in LMFDB) 2.64.x_jo $2$ (not in LMFDB) 2.64.b_aei $3$ (not in LMFDB) 2.64.abf_oe $6$ (not in LMFDB) 2.64.ab_aei $6$ (not in LMFDB) 2.64.bf_oe $6$ (not in LMFDB) 2.64.ap_ey $12$ (not in LMFDB) 2.64.p_ey $12$ (not in LMFDB)