No estoy encontrando un corto resumen de la composición de dos binarios cuadráticas formas, como el de Dirichlet. También, la edición de la Cox que tengo tiene una errata corregida en la segunda edición (2013), aquí está.
Dado $\gcd(a,a',B) = 1,$ definir
$$ X = xz-Cyw, $$
$$ Y = axw + a'yz + B yw, $$
$$ (a x^2 + B xy + a'C y^2) (a' z^2 + B zw + aC w^2) = aa'X^2 + B XY + C Y^2 $$
que usted debe comprobar!
Aquí están las formas binarias (primitiva) de discriminante $-284$
Discr -284 = 2^2 * 71 class number 7
all
284: < 1, 0, 71>
284: < 3, -2, 24>
284: < 3, 2, 24>
284: < 5, -4, 15>
284: < 5, 4, 15>
284: < 8, -2, 9>
284: < 8, 2, 9>
Los primeros números primos íntegramente representado por $3x^2 + 2xy+24y^2$
$$ 3, 29, 89, 103, 109, 151, 157, 191, $$ and below, we show how to represent each $p^7$ once we have $x,y.$
compared with $-71$ primitive, where this time a form represents the prime $2$
Discr -71 = 71 class number 7
all
71: < 1, 1, 18>
71: < 2, -1, 9>
71: < 2, 1, 9>
71: < 3, -1, 6>
71: < 3, 1, 6>
71: < 4, -3, 5>
71: < 4, 3, 5>
ummmm, $h(-71) = h(-284) = 7.$ Since $4 \cdot 3^7 - 284 = 92^2,$ the principal form is $\langle 1, 92, 2187\rangle.$ The class group is cyclic, everything is a power of $\langle 3, 92, 729 \rangle$ under Dirichlet's version of Gauss composition. All I am doing is repeatedly multiplying by $3 x^2 + 92 xy + 729 y^2,$ the rules for composition eventually give the quadratic form $\langle 2187, 92, 1 \rangle$ with variables which are homogeneous degree seven in the original $x,y.$ Oh, any form that represents $1$ is $SL_2 \mathbb Z$ equivalent to the principal form. At the very end, I show how to write $t^2 + 71 z^2 = (3 x^2 + 92 xy + 729 y^2)^7. $ Estoy mostrando el conjunto de la gp-pari sesión, no hay nada difícil una vez que conseguimos que la suerte de la expresión de los coeficientes de un generador del grupo.
a=3; a1=3; b=92; c=243; z = x; w = y;
zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? a=3; a1=3; b=92; c=243; z = x; w = y;
? zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? z
%3 = x^2 - 243*y^2
? w
%4 = 6*y*x + 92*y^2
?
a1 = 9; c = 81; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? a1 = 9; c = 81; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? z
%6 = x^3 - 729*y^2*x - 7452*y^3
? w
%7 = 27*y*x^2 + 828*y^2*x + 6277*y^3
?
a1 = 27; c = 27; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? a1 = 27; c = 27; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? z
%9 = x^4 - 1458*y^2*x^2 - 29808*y^3*x - 169479*y^4
? w
%10 = 108*y*x^3 + 4968*y^2*x^2 + 75324*y^3*x + 376280*y^4
?
?
a1 = 81; c = 9; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? a1 = 81; c = 9; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? z
%12 = x^5 - 2430*y^2*x^3 - 74520*y^3*x^2 - 847395*y^4*x - 3386520*y^5
? w
%13 = 405*y*x^4 + 24840*y^2*x^3 + 564930*y^3*x^2 + 5644200*y^4*x + 20889961*y^5
?
a1 = 243; c = 3; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? a1 = 243; c = 3; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
? z
%15 = x^6 - 3645*y^2*x^4 - 149040*y^3*x^3 - 2542185*y^4*x^2 - 20319120*y^5*x - 62669883*y^6
? w
%16 = 1458*y*x^5 + 111780*y^2*x^4 + 3389580*y^3*x^3 + 50797800*y^4*x^2 + 376019298*y^5*x + 1098952052*y^6
?
?
a1 = 729; c = 1; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
?
? a1 = 729; c = 1; zz = x * z - c * y * w ; ww = a * x * w + a1 * y * z + b * y * w; z = zz; w = ww;
?
? z
%18 = x^7 - 5103*y^2*x^5 - 260820*y^3*x^4 - 5931765*y^4*x^3 - 71116920*y^5*x^2 - 438689181*y^6*x - 1098952052*y^7
? w
%19 = 5103*y*x^6 + 469476*y^2*x^5 + 17795295*y^3*x^4 + 355584600*y^4*x^3 + 3948202629*y^5*x^2 + 23077993092*y^6*x + 55417244077*y^7
?
2187 * z^2 + 92 * z * w + w^2
( 3 * x^2 + 92 * x * y + 729 * y^2)^7
?
? 2187 * z^2 + 92 * z * w + w^2
%20 = 2187*x^14 + 469476*y*x^13 + 46911879*y^2*x^12 + 2892076488*y^3*x^11 + 122889105423*y^4*x^10 + 3807263630268*y^5*x^9 + 88688782583499*y^6*x^8 + 1578039270279536*y^7*x^7 + 21551374167790257*y^8*x^6 + 224815110103695132*y^9*x^5 + 1763324345027822661*y^10*x^4 + 10084047184857263688*y^11*x^3 + 39747900724268273397*y^12*x^2 + 96660945131267433924*y^13*x + 109418989131512359209*y^14
?
?
? ( 3 * x^2 + 92 * x * y + 729 * y^2)^7
%21 = 2187*x^14 + 469476*y*x^13 + 46911879*y^2*x^12 + 2892076488*y^3*x^11 + 122889105423*y^4*x^10 + 3807263630268*y^5*x^9 + 88688782583499*y^6*x^8 + 1578039270279536*y^7*x^7 + 21551374167790257*y^8*x^6 + 224815110103695132*y^9*x^5 + 1763324345027822661*y^10*x^4 + 10084047184857263688*y^11*x^3 + 39747900724268273397*y^12*x^2 + 96660945131267433924*y^13*x + 109418989131512359209*y^14
?
? 2187 * z^2 + 92 * z * w + w^2 - ( 3 * x^2 + 92 * x * y + 729 * y^2)^7
%22 = 0
?
t = w + 46 * z
t^2 + 71 * z^2
?
? t = w + 46 * z
%23 = 46*x^7 + 5103*y*x^6 + 234738*y^2*x^5 + 5797575*y^3*x^4 + 82723410*y^4*x^3 + 676824309*y^5*x^2 + 2898290766*y^6*x + 4865449685*y^7
?
?
?
? t^2 + 71 * z^2
%24 = 2187*x^14 + 469476*y*x^13 + 46911879*y^2*x^12 + 2892076488*y^3*x^11 + 122889105423*y^4*x^10 + 3807263630268*y^5*x^9 + 88688782583499*y^6*x^8 + 1578039270279536*y^7*x^7 + 21551374167790257*y^8*x^6 + 224815110103695132*y^9*x^5 + 1763324345027822661*y^10*x^4 + 10084047184857263688*y^11*x^3 + 39747900724268273397*y^12*x^2 + 96660945131267433924*y^13*x + 109418989131512359209*y^14
?
? t^2 + 71 * z^2 - ( 3 * x^2 + 92 * x * y + 729 * y^2)^7
%25 = 0
?
?
? t
%26 = 46*x^7 + 5103*y*x^6 + 234738*y^2*x^5 + 5797575*y^3*x^4 + 82723410*y^4*x^3 + 676824309*y^5*x^2 + 2898290766*y^6*x + 4865449685*y^7
?
? z
%27 = x^7 - 5103*y^2*x^5 - 260820*y^3*x^4 - 5931765*y^4*x^3 - 71116920*y^5*x^2 - 438689181*y^6*x - 1098952052*y^7
?
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