weil: Name orbits in `DS-get-orbits'.

Note that the numbering (the letters) depend on the smith basis.

* src/weil.cpp (base26): New function.
(ds_info): Output orbit name for `Pretty' and 'LaTeX' output formats.

src/weil.cpp

@@ -3,6 +3,7 @@
 
 #include <algorithm>
 #include <cstdlib>
+#include <cstring>
 #include <exception>
 #include <fstream>
 #include <iostream>
@@ -272,6 +273,26 @@ lattice_smith_gram_matrix(int argc, const char** argv, weil::IOFormat style)
 	return EXIT_SUCCESS;
 }
 
+std::string
+base26(uint64_t n)
+{
+	const char* alphabet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
+	const uint64_t len = strlen(alphabet) - 1;
+
+	if (n == 0) {
+		return { alphabet[0] };
+	}
+
+	std::string s;
+	while (n > 0) {
+		s.push_back(alphabet[n % len]);
+		n /= len;
+	}
+
+	std::ranges::reverse(s);
+	return s;
+}
+
 int
 ds_get_orbits(int argc, const char** argv, weil::IOFormat style)
 {
@@ -307,7 +328,7 @@ ds_get_orbits(int argc, const char** argv, weil::IOFormat style)
 	switch (style) {
 	using enum weil::IOFormat;
 	case Pretty:
-		start = "representative\tlength\tnorm\torder\n";
+		start = "representative\tname\tlength\tnorm\torder\n";
 		stop = "";
 		prefix = "";
 		infix = "\t";
@@ -331,7 +352,7 @@ ds_get_orbits(int argc, const char** argv, weil::IOFormat style)
 		sep = ",\\\n";
 		break;
 	case LaTeX:
-		start = "\\begin{tabular}{c c c c}\n";
+		start = "\\begin{tabular}{c c c c c}\n";
 		stop = "\n\\end{tabular}";
 		prefix = "$";
 		infix = "$ & $";
@@ -342,6 +363,8 @@ ds_get_orbits(int argc, const char** argv, weil::IOFormat style)
 
 	std::cout << start;
 	bool first{ true };
+	uint64_t offset_idx{ 0 };
+	mpz_class last_offset{ 0_mpz };
 	for (auto&& o : orbits) {
 		if (first) {
 			first = false;
@@ -349,15 +372,27 @@ ds_get_orbits(int argc, const char** argv, weil::IOFormat style)
 			std::cout << sep;
 		}
 
+		mpz_class offset{ (D.level() * D.q(o.representative)) };
+		if (last_offset != offset) {
+			last_offset = offset;
+			offset_idx = 0;
+		}
+
 		std::cout << prefix;
 		format_eigen(std::cout, o.representative, style);
 		std::cout << infix;
+		if (style == weil::IOFormat::Pretty || style == weil::IOFormat::LaTeX) {
+			std::cout << offset << base26(offset_idx);
+			std::cout << infix;
+		}
 		std::cout << o.length;
 		std::cout << infix;
 		std::cout << D.q(o.representative);
 		std::cout << infix;
 		std::cout << D.element_order(o.representative);
 		std::cout << suffix;
+
+		offset_idx++;
 	}
 	std::cout << stop << std::endl;