# Properties

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

# Learn more about

## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $1 - 25 x + 280 x^{2} - 1600 x^{3} + 4096 x^{4}$ Frobenius angles: $\pm0.136006851804$, $\pm0.273761000528$ Angle rank: $2$ (numerical) Number field: 4.0.29189.1 Galois group: $D_{4}$ Jacobians: 6

This isogeny class is simple and geometrically 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2752 16517504 68870481472 281625326195456 1152987408580898752 4722381057964796230784 19342813538567409218189632 79228163789827253090325440000 324518556796747096480989254402752 1329227998519869554003002657855557504

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 40 4032 262720 16786176 1073803200 68719688832 4398046607680 281474981242368 18014398683693760 1152921506979041152

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The endomorphism algebra of this simple isogeny class is 4.0.29189.1.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

## 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 2.64.z_ku $2$ (not in LMFDB)