We present an explicit one-parameter family of elliptic curves defined over ℚ(t) possessing at least four independent rational points, where the fourth point is defined over a quadratic extension. By specializing at t0= −842/35, we compute the Néron-Tate height pairing matrix of these points numerically using SageMath, establishing their linear independence and hence the curve’s rank of at least 4. This construction builds upon interpolation techniques and explicit extension field definitions, contributing a concrete example in the study of high rank elliptic curve families.
| Published in | American Journal of Applied Mathematics (Volume 13, Issue 5) |
| DOI | 10.11648/j.ajam.20251305.14 |
| Page(s) | 344-347 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Elliptic Curves, Mordell-Weil Rank, Parametric Families, Quadratic Extensions, Rational Points, Polynomial Interpolation, Specialization, Néron-Tate Height
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APA Style
Vincent, K. K. (2025). Explicit Construction of a Parametric Family of Elliptic Curves of Rank 4 via a Quadratic Extension. American Journal of Applied Mathematics, 13(5), 344-347. https://doi.org/10.11648/j.ajam.20251305.14
ACS Style
Vincent, K. K. Explicit Construction of a Parametric Family of Elliptic Curves of Rank 4 via a Quadratic Extension. Am. J. Appl. Math. 2025, 13(5), 344-347. doi: 10.11648/j.ajam.20251305.14
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Download@article{10.11648/j.ajam.20251305.14,
author = {Kouakou Kouassi Vincent},
title = {Explicit Construction of a Parametric Family of Elliptic Curves of Rank 4 via a Quadratic Extension
},
journal = {American Journal of Applied Mathematics},
volume = {13},
number = {5},
pages = {344-347},
doi = {10.11648/j.ajam.20251305.14},
url = {https://doi.org/10.11648/j.ajam.20251305.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251305.14},
abstract = {We present an explicit one-parameter family of elliptic curves defined over ℚ(t) possessing at least four independent rational points, where the fourth point is defined over a quadratic extension. By specializing at t0= −842/35, we compute the Néron-Tate height pairing matrix of these points numerically using SageMath, establishing their linear independence and hence the curve’s rank of at least 4. This construction builds upon interpolation techniques and explicit extension field definitions, contributing a concrete example in the study of high rank elliptic curve families.
},
year = {2025}
}
TY - JOUR T1 - Explicit Construction of a Parametric Family of Elliptic Curves of Rank 4 via a Quadratic Extension AU - Kouakou Kouassi Vincent Y1 - 2025/10/22 PY - 2025 N1 - https://doi.org/10.11648/j.ajam.20251305.14 DO - 10.11648/j.ajam.20251305.14 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 344 EP - 347 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20251305.14 AB - We present an explicit one-parameter family of elliptic curves defined over ℚ(t) possessing at least four independent rational points, where the fourth point is defined over a quadratic extension. By specializing at t0= −842/35, we compute the Néron-Tate height pairing matrix of these points numerically using SageMath, establishing their linear independence and hence the curve’s rank of at least 4. This construction builds upon interpolation techniques and explicit extension field definitions, contributing a concrete example in the study of high rank elliptic curve families. VL - 13 IS - 5 ER -
Applied Fondamental Sciences Department, Nangui Abrogoua University, Abidjan, Côte d’Ivoire
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