Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields

DOI10.1112/S1461157015000157zbMath1397.11089arXiv1406.0655OpenAlexW3098171681MaRDI QIDQ2942125

Peter Bruin, Filip Najman

Publication date: 26 August 2015

Published in: LMS Journal of Computation and Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1406.0655

zbMATH Keywords

elliptic curvescyclic kernelisogenyquadratic points

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Arithmetic aspects of modular and Shimura varieties (11G18) Software, source code, etc. for problems pertaining to number theory (11-04) Global ground fields in algebraic geometry (14G25)

Related Items (17)

Computing representation matrices for the action of Frobenius on cohomology groups ⋮ Quadratic points on bielliptic modular curves ⋮ Splitting of primes in number fields generated by points on some modular curves ⋮ Quadratic points on non-split Cartan modular curves ⋮ Fermat’s Last Theorem and modular curves over real quadratic fields ⋮ On elliptic curves with p-isogenies over quadratic fields ⋮ COMPUTING POINTS ON BIELLIPTIC MODULAR CURVES OVER FIXED QUADRATIC FIELDS ⋮ Computing quadratic points on modular curves 𝑋₀(𝑁) ⋮ Quadratic points on modular curves with infinite Mordell–Weil group ⋮ Quadratic points on modular curves ⋮ \(\mathbb{Q}\)-curves over odd degree number fields ⋮ The twisting Sato-Tate group of the curve \(y^2 = x^{8} - 14x^4 + 1\) ⋮ Elliptic curves over real quadratic fields are modular ⋮ Computing models for quotients of modular curves ⋮ Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields ⋮ Rational points on twisted modular curves ⋮ On the level of modular curves that give rise to isolated \(j\)-invariants

Uses Software

• Magma

Cites Work

• On the number of Q-isomorphism classes of elliptic curves in each Q- isogeny class
• Equations of hyperelliptic modular curves
• Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
• Modular curves and the Eisenstein ideal
• Bielliptic modular curves
• Torsion points on elliptic curves and \(q\)-coefficients of modular forms
• Bounds for the torsion of elliptic curves over number fields
• Implementing 2-descent for Jacobians of hyperelliptic curves
• Torsion points on elliptic curves defined over quadratic fields
• On the Modular Curves X ₀ (125), X ₁ (25), X ₁ (49)
• The Modular Curve X ₀ (169) and Rational Isogeny
• The modular curves X₀(65) and X₀(91) and rational isogeny
• The modular curve X₀(39) and rational isogeny
• On the torsion of elliptic curves over cubic number fields
• ON THE TORSION OF ELLIPTIC CURVES OVER QUARTIC NUMBER FIELDS
• Isogenies of polyquadratic \(\mathbb Q\)-curves to their Galois conjugates
• Hyperelliptic modular curves

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