Class number calculation of a sextic field from the elliptic unit

Improvement of Nakamula’s upper bound for the absolute discriminant of a sextic number field with two real conjugates

“…To be more precise, on p. 231 of [2], the constant term in the definition of A?, should be -104 instead of -108 , and the coefficient of y~3 in the closing line should be +1280 instead of -1280. Hence, solving F'(x) = 0 analytically-in order to find stationary points-is either trivial (which is not the case here) or impossible, the more so because a parameter a is involved.…”

“…In his beautiful paper [2], Nakamula shows great arithmetic skills in the way he calculates fundamental units and class numbers of number fields of absolute degree 6 over Q. In the paper cited, Nakamula considers sextic fields K with a real quadratic subfield K2 and a complex cubic subfield K3.…”

“…To ensure that e generates H, is is sufficient to check that a relation of the form (2) e = £," for a £ e H and n e N with « > 2 is impossible. To ensure that e generates H, is is sufficient to check that a relation of the form (2) e = £," for a £ e H and n e N with « > 2 is impossible.…”

“…To ensure that e generates H, is is sufficient to check that a relation of the form (2) e = £," for a £ e H and n e N with « > 2 is impossible. Let e and n be as in (2). If the inverse T~l of r can be explicitly evaluated at each relevant value, then a useful upper bound for the exponent n in (2) can be obtained as follows.…”

“…If the inverse T~l of r can be explicitly evaluated at each relevant value, then a useful upper bound for the exponent n in (2) can be obtained as follows. Let e and n be as in (2). Let e and n be as in (2).…”