History of Hilbert class polynomials $H_\Delta$

   '-163':     number: x + 262537412640768000-    comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-163})$; $640320^3=-H_\Delta(0)$-      is the integer that HREF{Ramanujan_constant}[Ramanujan's constant $e^{\pi\sqrt{163}}$]-      falls $744$ short of+    comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-163})$; $H_\Delta(0)=640320^3$,+      the integer that HREF{Ramanujan_constant}[Ramanujan's constant $e^{\pi\sqrt{163}}$]+      exceeds by $744$ to within $10^{-12}$; the root $j\bigl(\tfrac{1+\sqrt{-163}}{2}\bigr)=-640320^3$   '-164':     number: x^8 - 296853791160440320*x^7 - 161936389233870440957755392*x^6 - 107971538556531472065498397540352*x^5

 Comments:   comment-size: The coefficients grow like $\exp\bigl(\pi\sqrt{|\Delta|}\sum 1/a\bigr)$,-    the sum over the reduced forms, and the degree like $\sqrt{|\Delta|}$; the longest-    entry here is $\Delta=-239$, of degree $15$ and $1106$ characters. The OEIS entry-    CITE{OEIS} carries the coefficients to $\Delta=-500$, and the Sage and PARI programs-    below give any further one.+    the sum over the reduced forms, and the degree like $\sqrt{|\Delta|}$. The OEIS+    entry CITE{OEIS} carries the coefficients to $\Delta=-500$, and the Sage and PARI+    programs here give any further one.   comment-roots: The roots of $H_\Delta$ are the singular moduli of discriminant $\Delta$,     the values $j(\tau)$ at the imaginary quadratic $\tau$ with $\mathrm{End}\langle\tau,1\rangle=\mathcal{O}_\Delta$;

     class field of $\mathcal{O}_\Delta$ over $K$, which is the Hilbert class field     of $K$ exactly when $f=1$. Each entry's comment names the order and its class-    number $h(\Delta)$, which is the degree. Of the 150 entries, 94 are fundamental-    and 56 are not.+    number $h(\Delta)$, which is the degree.   comment-cm-method: 'This is the polynomial the CM method reduces modulo $p$: for     a prime $p$ with $4p=X^2-\Delta Y^2$, the roots of $H_\Delta$ in $\mathbb{F}_p$

   formula-cube: '$H_\Delta(0)$ is a perfect cube whenever $3\nmid\Delta$: then Weber''s     $\gamma_2=\sqrt[3]{j}$ takes values in the ring class field, so the norm of $j$-    is a cube (CITE{Cox}, §12). Checked on every entry.'+    is a cube (CITE{Cox}, §12).'   formula-forms: '$H_\Delta(x)=\prod_{(a,b,c)}\Bigl(x-j\bigl(\tfrac{-b+\sqrt{\Delta}}{2a}\bigr)\Bigr)$,     the product over the reduced primitive positive definite binary quadratic forms

     and $w_\Delta$ the number of units of $\mathcal{O}_\Delta$; only $\Delta=(t^2-4\ell)/f^2$     with $t^2<4\ell$ occur, and $\sum_\Delta r_\ell(\Delta)\,h(\Delta)=2\ell$ (CITE{Cox},-    §13). For $\ell=2$: $\Phi_2(x,x)=-(x-1728)(x+3375)^2(x-8000)$. Checked against-    the stored $\Phi_\ell$ for $\ell=2,3,5,7,11$.'+    §13). For $\ell=2$: $\Phi_2(x,x)=-(x-1728)(x+3375)^2(x-8000)$.'   formula-splitting: 'For a prime $p\nmid\Delta$: $4p=X^2-\Delta Y^2$ has a solution     in integers if and only if $\left(\frac{\Delta}{p}\right)=1$ and $H_\Delta$ has-    a root modulo $p$ (CITE{Cox}, Theorem 9.2, stated there for $\Delta=-4n$). Checked-    here for every $\Delta$ in the table and every prime $p<400$, both directions,-    with no exception.'+    a root modulo $p$ (CITE{Cox}, Theorem 9.2, stated there for $\Delta=-4n$ and for+    $p\nmid\operatorname{disc}H_\Delta$, a hypothesis that is automatic once $p$ splits+    in $K$: two singular moduli of discriminant $\Delta$ coincide modulo $p$ only+    at supersingular reduction).' Programs:   program-pari:

   relation: $\Phi_\ell(x,x)$ is a product of powers of $H_\Delta$ - table: HREF{Ramanujan_constant}[Ramanujan's constant]-  relation: $e^{\pi\sqrt{163}}$ is within $10^{-12}$ of $744-H_{-163}(0)=640320^3+744$+  relation: $e^{\pi\sqrt{163}}$ is within $10^{-12}$ of $H_{-163}(0)+744=640320^3+744$ - table: HREF{Q-expansion_of_the_j-invariant}[$q$-expansion of the $j$-invariant]   relation: the function whose values at imaginary quadratic points are the roots

     in balls of radius below $1/2$ by the product $\prod(x-j(\tau))$ over those forms     computed in ComplexBallField at 1500 bits, which determines an integer coefficient.-    All 150 were also compared with the 250 rows of the OEIS b-file of A305474.-  complete-note: every such discriminant with $|\Delta|\leq 300$ is here+    All 150 were also compared with the 250 rows of the OEIS b-file of A305474. The+    stored polynomials were also checked against the cube formula on every entry,+    against the diagonal HREF{Modular_polynomials_for_j-invariant}[of the modular+    polynomials for $j$] for $\ell=2,3,5,7,11$, and against the splitting criterion+    for every $\Delta$ here and every prime $p<400$, in both directions.+  complete-note: every such discriminant with $|\Delta|\leq 300$ is here, 94 fundamental+    and 56 not; the range is set by how long an entry becomes Display properties:   number-header: $H_\Delta(x)$