N \equiv 3 mod 8新类不变量在奇异值k_N处的根解
Solutions by radicals at singular values k_N from new class invariants for N \equiv 3 mod 8
摘要：
对于平方自由$N\equiv3$模8和$N$互质3，我展示了如何减少奇异值$k_N$的根，使用一种新的对$[f,g]$的实数是代数整数的希尔伯特类字段的$Q(\sqrt{-N})$。一个是模级48的类不变式，具有增长$g=α(N)\exp(π\sqrt{N}/48)+o(1)$，其中$α(N)\in[-\sqrt2,\sqrt2]$由$N$模64的余数唯一确定。因此PDMATHAMT是一个非常经济的类域生成器。对于素数$N\equiv3$ mod 4，我猜想Chowla-Selberg公式提供了类域的一个代数{ \em单位}，并确定了具有$N<2000$的155种情形的它的最小多项式.对于N=2317723，类号为$h(-N)=105$，我在90毫秒内计算出$g$的最小多项式。它的高度小于{ \em cube}根的生成多项式的高度由双η商方法{ \em Pari-GP}。我减少了完整的椭圆积分$K_{2317723}$的$Γ$功能的自由基和值，通过确定Chowla-Selberg单位和解决的腐败，五次和三次方程，产生子字段的类字段。我的结论是，剩余3模8，最初丢弃在椭圆曲线素性证明，优于剩余7。
英文摘要：
For square-free $N\equiv3$ mod 8 and $N$ coprime to 3, I show how to reduce the singular value $k_N$ to radicals, using a novel pair $[f,g]$ of real numbers that are algebraic integers of the Hilbert class field of $Q(\sqrt{-N})$. One is a class invariant of modular level 48, with a growth $g=α(N)\exp(π\sqrt{N}/48)+o(1)$, where $α(N)\in[-\sqrt2,\sqrt2]$ is uniquely determined by the residue of $N$ modulo 64. Hence $g$ is a very economical generator of the class field. For prime $N\equiv3$ mod 4, I conjecture that the Chowla--Selberg formula provides an algebraic {\em unit} of the class field and determine its minimal polynomial for the 155 cases with $N<2000$. For N=2317723, with class number $h(-N)=105$, I compute the minimal polynomial of $g$ in 90 milliseconds. Its height is smaller than the {\em cube} root of the height of the generating polynomial found by the double eta-quotient method of {\em Pari-GP}. I reduce the complete elliptic integral $K_{2317723}$ to radicals and values of the $Γ$ function, by determining the Chowla--Selberg unit and solving the septic, quintic and cubic equations that generate sub-fields of the class field. I conclude that the residue 3 modulo 8, initially discarded in elliptic curve primality proving, outperforms the residue 7.
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