A family of monogenic S4 quartic fields arising from elliptic curves.
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| Title: | A family of monogenic S4 quartic fields arising from elliptic curves. |
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| Authors: | Gassert, T. Alden1 gassert@hws.edu, Smith, Hanson1,2 hanson.smith@colorado.edu, Stange, Katherine E.2 kstange@math.colorado.edu |
| Source: | Journal of Number Theory. Apr2019, Vol. 197, p361-382. 22p. |
| Subjects: | Polynomials, Elliptic curves, Integers, Foundations of arithmetic, Algorithms |
| Abstract: | Abstract We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial 3-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic S 4 fields given by T 4 − 6 T 2 − α T − 3 for α ∈ Z such that α ± 8 are squarefree, are monogenic. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 133557266 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A family of monogenic S4 quartic fields arising from elliptic curves. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Gassert%2C+T%2E+Alden%22">Gassert, T. Alden</searchLink><relatesTo>1</relatesTo><i> gassert@hws.edu</i><br /><searchLink fieldCode="AR" term="%22Smith%2C+Hanson%22">Smith, Hanson</searchLink><relatesTo>1,2</relatesTo><i> hanson.smith@colorado.edu</i><br /><searchLink fieldCode="AR" term="%22Stange%2C+Katherine+E%2E%22">Stange, Katherine E.</searchLink><relatesTo>2</relatesTo><i> kstange@math.colorado.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Apr2019, Vol. 197, p361-382. 22p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Elliptic+curves%22">Elliptic curves</searchLink><br /><searchLink fieldCode="DE" term="%22Integers%22">Integers</searchLink><br /><searchLink fieldCode="DE" term="%22Foundations+of+arithmetic%22">Foundations of arithmetic</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Abstract We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial 3-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic S 4 fields given by T 4 − 6 T 2 − α T − 3 for α ∈ Z such that α ± 8 are squarefree, are monogenic. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jnt.2018.09.026 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 22 StartPage: 361 Subjects: – SubjectFull: Polynomials Type: general – SubjectFull: Elliptic curves Type: general – SubjectFull: Integers Type: general – SubjectFull: Foundations of arithmetic Type: general – SubjectFull: Algorithms Type: general Titles: – TitleFull: A family of monogenic S4 quartic fields arising from elliptic curves. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Gassert, T. Alden – PersonEntity: Name: NameFull: Smith, Hanson – PersonEntity: Name: NameFull: Stange, Katherine E. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Text: Apr2019 Type: published Y: 2019 Identifiers: – Type: issn-print Value: 0022314X Numbering: – Type: volume Value: 197 Titles: – TitleFull: Journal of Number Theory Type: main |
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