# Properties

 Label 2.64.ax_jw 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: $1 - 23 x + 256 x^{2} - 1472 x^{3} + 4096 x^{4}$ Frobenius angles: $\pm0.178048525745$, $\pm0.299165991752$ Angle rank: $2$ (numerical) Number field: 4.0.54332.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2858 16713584 69003486266 281672058949088 1152988655399793738 4722372874184682990416 19342808661288122194103642 79228161630532184254744369088 324518554878270321786318198751082 1329227996349844068896785019549565744

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 42 4080 263226 16788960 1073804362 68719569744 4398045498714 281474973571008 18014398577196906 1152921505096844080

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The endomorphism algebra of this simple isogeny class is 4.0.54332.1.
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.x_jw $2$ (not in LMFDB)