History of Weber class polynomials $W_D$

 Title: Weber class polynomials $W_D$ Definition: For $D<0$, $D\equiv1\pmod 8$ and $3\nmid D$, put $m=-D$. Let $\omega_D=\mathfrak-  f(i\sqrt m)/\sqrt2$, with $\mathfrak f$ Weber's function CITE{Schertz2002}. Listed-  is the monic minimal polynomial $W_D(x)\in\mathbb Z[x]$ of $\omega_D$.+  f(i\sqrt m)/\sqrt2$, with $\mathfrak f$ Weber's function CITE{Schertz2002} CITE{YuiZagier1997}.+  $W_D(x)\in\mathbb Z[x]$ is the minimal polynomial of $\omega_D$. Parameters:   D:

     constraints: $D<0$, $D\equiv1\pmod 8$, and $3\nmid D$ Comments:-  comment-discriminant: The point $i\sqrt m$ has discriminant $4D$, while the polynomial-    is indexed by the discriminant $D$ of the order whose ring class field is generated-    by the Weber class invariant. This is why the parameter is $D$, not $4D$.+  comment-discriminant: The order $\mathbb Z[i\sqrt m]$ has discriminant $4D$, while+    the polynomial is indexed by the discriminant $D$ of the order whose ring class+    field is generated by the Weber class invariant CITE{Cox}. This is why the parameter+    is $D$, not $4D$.   comment-entries: Each entry's comment gives the class number $h(D)$, which is the     degree of $W_D$, and says whether $D$ is fundamental.

     method because Weber class invariants have much smaller height than the corresponding     singular moduli.+  comment-pari: For the entries here, PARI's `polclass(D,1)` CITE{PARIpolclass} is+    the minimal polynomial of $1/\omega_D$ or $-1/\omega_D$; the sign depends on $D$.+    For example, `polclass(-23,1)` returns $x^3-x^2+1$, which corresponds here to+    $W_{-23}(x)=x^3-x-1$. Formulas:   formula-eta: $\mathfrak f(\tau)=\eta(\tau)^2/(\eta(\tau/2)\eta(2\tau))$, where $\eta$

   program-pari:     language: PARI/GP-    code: 'P = polclass(-23, 1)      \\ PARI''s Weber class-invariant polynomial--      Pol(Vec(P))               \\ x^3 - x^2 + 1; W_{-23}(x)=x^3-x-1 after the reciprocal/sign-      transform'+    code: '\p 100++      D = -23;++      P = polclass(D, 1);++      w = weber(I*sqrt(-D))/sqrt(2);++      Q = polrecip(P);++      S = polrecip(subst(P, x, -x));++      if (abs(subst(S, x, w)) < abs(subst(Q, x, w)), Q = S);++      Q / pollead(Q)'   program-sage:     language: Sage

       from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing +      from sage.rings.real_mpfr import RealField++      from sage.rings.complex_mpfr import ComplexField++      from sage.libs.pari import pari+       from sage.schemes.elliptic_curves.cm import hilbert_class_polynomial 

       H = hilbert_class_polynomial(4*D).change_ring(ZZ) -      relation = (256*x^24 - 1)^3 - j*x^24--      factor(H(j).resultant(relation, j))'+      relation = (256*x**24 - 1)**3 - j*x**24++      factors = H(j).resultant(relation, j).factor()++      w = ComplexField(200)(pari(''weber(I*sqrt(%d))'' % -D)) / RealField(200)(2).sqrt()++      min((p.univariate_polynomial() for p, e in factors), key=lambda p: abs(p(w)))' Similar tables: - table: HREF{Hilbert_class_polynomials}[Hilbert class polynomials $H_\Delta$]

 - table: HREF{Ramanujan's_class_invariants_G_n_with_Weber's_mathfrak_f_sqrt-n}[Ramanujan's     class invariants $G_n$, with Weber's $\mathfrak f(\sqrt{-n})$]-  relation: stores individual Weber values at $i\sqrt n$; this table stores the minimal-    polynomial when $n=-D$ and the Weber value is a class invariant+  relation: stores $G_n$ and $\mathfrak f(i\sqrt n)$; for $n=-D$, the roots of $W_D$+    include $\omega_D=2^{-1/4}G_n$ - table: HREF{Modular_polynomials_for_Weber's_f-function}[Modular polynomials for     Weber's $f$-function]