# Properties

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

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## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ Weil polynomial: $( 1 - 13 x + 64 x^{2} )( 1 + 13 x + 64 x^{2} )$ Frobenius angles: $\pm0.198106042756$, $\pm0.801893957244$ Angle rank: $1$ (numerical) Jacobians: 426

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 426 curves, and hence is principally polarizable:

• $y^2+(x^2+x)y=ax^5+(a^3+a^2+a)x^3+(a^2+1)x^2+(a^3+1)x$
• $y^2+(x^2+x)y=ax^5+(a^5+a^3+a^2+a)x^4+(a^3+a^2+a)x^3+(a^5+a^3+a+1)x^2+(a^3+1)x$
• $y^2+(x^2+x)y=(a^5+a^4+a^2+a+1)x^5+(a^3+a^2+a+1)x^3+(a^5+a^3+a)x^2+(a^4+a)x$
• $y^2+(x^2+x)y=(a^5+a^4+a^2+a+1)x^5+(a^4+a^3+1)x^4+(a^3+a^2+a+1)x^3+(a^5+a^4+a+1)x^2+(a^4+a)x$
• $y^2+(x^2+x)y=(a^4+a+1)x^5+(a^3+a^2+a)x^3+(a^5+a^4+a+1)x^2+(a^5+a^3+a^2+a)x$
• $y^2+(x^2+x)y=(a^4+a+1)x^5+(a^5+a^3+a^2)x^4+(a^3+a^2+a)x^3+(a^4+a^3+a^2+a+1)x^2+(a^5+a^3+a^2+a)x$
• $y^2+(x^2+x)y=a^2x^5+(a^3+a^2+a+1)x^3+(a^4+1)x^2+(a^4+a^3+a)x$
• $y^2+(x^2+x)y=a^2x^5+(a^5+a^3+1)x^4+(a^3+a^2+a+1)x^3+(a^5+a^4+a^3)x^2+(a^4+a^3+a)x$
• $y^2+(x^2+x)y=a^4x^5+(a^3+a^2+a)x^3+(a^5+a^3)x^2+(a^5+a^4+a^2+a)x$
• $y^2+(x^2+x)y=a^4x^5+(a^5+a^4+a^3+a+1)x^4+(a^3+a^2+a)x^3+(a^4+a+1)x^2+(a^5+a^4+a^2+a)x$
• $y^2+(x^2+x)y=(a^5+a^4+a)x^5+(a^3+a^2+a+1)x^3+(a+1)x^2+(a^5+a^4+a^3+a^2+a)x$
• $y^2+(x^2+x)y=(a^5+a^4+a)x^5+(a^4+a^3+a^2+1)x^4+(a^3+a^2+a+1)x^3+(a^4+a^3+a^2+a)x^2+(a^5+a^4+a^3+a^2+a)x$
• $y^2+(x^2+x+a^5+a^3+a)y=(a+1)x^5+(a+1)x^3+a^4x^2+(a^5+a^4+a^3+1)x+a^5+a+1$
• $y^2+(x^2+x+a^5+a^3+a)y=(a+1)x^5+(a^5+a^3+a)x^4+(a+1)x^3+(a^5+a^4+a^3+a)x^2+(a^5+a^4+a^3+1)x+a^4+a+1$
• $y^2+(x^2+x+a^5+a^3+a^2+a)y=(a^4+a^2+1)x^5+(a^5+a^4)x^3+(a^3+a)x+a^3+a+1$
• $y^2+(x^2+x+a^5+a^3+a^2+a)y=(a^4+a^2+1)x^5+(a^5+a^4+a^3+a)x^4+(a^5+a^4)x^3+(a^5+a^4+a^3+a)x^2+(a^3+a)x+a^4+a^2+a+1$
• $y^2+(x^2+x)y=(a^5+a^2+1)x^5+(a^5+a^4+a^3+a+1)x^3+(a^3+a+1)x^2+(a^4+a^2+1)x$
• $y^2+(x^2+x)y=(a^5+a^2+1)x^5+(a^3+a)x^4+(a^5+a^4+a^3+a+1)x^3+x^2+(a^4+a^2+1)x$
• $y^2+(x^2+x+a^5+a^4+a^3+a+1)y=(a^2+1)x^5+(a^2+1)x^3+(a^5+a^4+a^2+a+1)x^2+(a^4+a^2)x+a^4+a^3+a+1$
• $y^2+(x^2+x+a^5+a^4+a^3+a+1)y=(a^2+1)x^5+(a^5+a^4+a^3+a+1)x^4+(a^2+1)x^3+(a^3+a^2)x^2+(a^4+a^2)x+a^4+a^3+a^2+1$
• and 406 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4056 16451136 68719911624 281693525577984 1152921502463163576 4722426253610370317376 19342813113840728124788136 79228157538525417810440193024 324518553658426719376279169811864 1329227990841918495641742802133107776

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 65 4015 262145 16790239 1073741825 68720346511 4398046511105 281474959033279 18014398509481985 1152921500319480175

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The isogeny class factors as 1.64.an $\times$ 1.64.n 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 $\F_{2^{12}}$ is 1.4096.abp 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-87})$$$)$
All geometric endomorphisms are defined over $\F_{2^{12}}$.

## 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.aba_ll $2$ (not in LMFDB) 2.64.ba_ll $2$ (not in LMFDB) 2.64.a_bp $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.aba_ll $2$ (not in LMFDB) 2.64.ba_ll $2$ (not in LMFDB) 2.64.a_bp $4$ (not in LMFDB) 2.64.an_eb $6$ (not in LMFDB) 2.64.n_eb $6$ (not in LMFDB)