NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5^A

Abstract

Abstract. We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form P SL 2 (5) 5 .10, and field discriminant 5 69 . A closely related polynomial has degree 120, Galois group of the form SL 2 (5) 5 .20, and field discriminant 5 311 . We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5 −1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5 4 .

On computing Belyi maps

“…Further applications of the explicit study of Belyȋ maps have been found in inverse Galois theory, specifically the regular realization of Galois groups over small number fields: see the tomes of Matzat [120], Malle-Matzat [107], and Jensen-Ledet-Yui [80]. Upon specialization, one obtains Galois number fields with small ramification set: Roberts [131,132,133], Malle-Roberts [112], and Jones-Roberts [86] have used the specialization of three-point covers to exhibit number fields with small ramification set or root discriminant. The covering curves obtained are often interesting in their own right, spurring further investigation in the study of low genus curves (e.g., the decomposition of their Jacobian [127]).…”

Explicit Methods for Hilbert Modular Forms

“…and minimal conductor N. Then N = p 5 p 7 , where p 5 = ((−2b 3 − 12b 2 + 31b + 25)/7) is the unique prime above 5 and p 7 = ((−2b 4 − 9b 3 + 8b 2 + 53b + 6)/7) is one of the five primes above 7. Roberts [50] showed that the mod 5 Galois representation…”

“…The elliptic curve E found by Roberts [50] was obtained from our computations of Hilbert modular forms at level p 5 over F . The space S 2 (p 5 ) new has 2 Hecke constituents of dimension 10 and 20, respectively.…”

“…Roberts [50] has given an explicit equation for the number field L (obtained from the 5-division polynomial of the elliptic curve E): the field L is the the splitting field of the polynomial…”