A database of number fields

Jones, John W.; Roberts, David P.

doi:10.1112/s1461157014000424

Cited by 53 publications

(87 citation statements)

References 30 publications

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“…We know from [10] that this set consists of 2080 fields. We list these fields by increasing discriminant, K 1 , …, K 2080 , with the resolution of ties conveniently not affecting the explicit results appearing in Table 3.…”

Section: Transferring Completeness Resultsmentioning

confidence: 99%

“…Early work for quartics, quintics, sextics, and septics include respectively [2][3][4]7,13,[21][22][23]25,31]. Further results towards 2 in higher degrees are extractable from the websites associated to [10,12,14].…”

Section: Overviewmentioning

confidence: 99%

“…The fourth entry gives the position of the source number field on the complete list ordered by Galois root discriminant. This information lets readers obtain further information from [10], such as a defining polynomial and details on ramification.…”

Section: First Four Columnsmentioning

confidence: 99%

“…This method, applied to both old and newer lists presented in [10], accounts for all but one of the δ 1 reported in Roman type on the same line as a 0 in the # column. The remaining case of an established δ 1 is for the type (GL 3 (2), χ 7 ).…”

Section: Known and Unknown Minimal Root Conductorsmentioning

confidence: 99%

“…However examples like the one in the previous paragraph suggest to us that in many cases the resulting increase in d towards δ 1 would be very small. Tables 3, 4, 5, 6 and 7 make implicit reference to many Galois number fields, and all necessary complete lists are accessible from the database [10]. Table 8 presents a small excerpt from this database by giving six polynomials f (x).…”

Section: The Ratio δ 1 /Dmentioning

confidence: 99%

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Artin L-functions of small conductor

Jones

Roberts

2017

Res. number theory

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Section: Transferring Completeness Resultsmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: First Four Columnsmentioning

confidence: 99%

Section: Known and Unknown Minimal Root Conductorsmentioning

confidence: 99%

Section: The Ratio δ 1 /Dmentioning

confidence: 99%

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Artin L-functions of small conductor

Jones

Roberts

2017

Res. number theory

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Division Polynomials with Galois Group $$SU_{3}(3).2\mathop{\cong}G_{2}(2)$$

Roberts

2015

Fields Institute Communications

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We use a rigidity argument to prove the existence of two related degree twenty-eight covers of the projective plane with Galois group SU 3 (3).2 ∼ = G 2 (2). Constructing corresponding two-parameter polynomials directly from the defining group-theoretic data seems beyond feasablity. Instead we provide two independent constructions of these polynomials, one from 3-division points on covers of the projective line studied by Deligne and Mostow, and one from 2-division points of genus three curves studied by Shioda. We explain how one of the covers also arises as a 2-division polynomial for a family of G 2 motives in the classification of Dettweiler and Reiter. We conclude by specializing our two covers to get interesting three-point covers and number fields which would be hard to construct directly.

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