# Reduction of Binary Cubic and Quartic Forms

@article{Cremona1999ReductionOB, title={Reduction of Binary Cubic and Quartic Forms}, author={John Cremona}, journal={Lms Journal of Computation and Mathematics}, year={1999}, volume={2}, pages={62-92} }

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over the set of rational numbers. Remarks are given concerning the extension of these results to forms defined over number fields…

## 84 Citations

### On the reduction theory of binary forms

- Mathematics, Computer Science
- 2003

A reduction theory for binary forms of degree 3 and 4 with integer coefficients for elliptic curves over Q is extended to forms of higher degree, one application of this is to the study of hyperelliptic curves.

### On binary cubic and quartic forms

- MathematicsJournal de Théorie des Nombres de Bordeaux
- 2019

In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic…

### Reduction theory of binary forms

- Mathematics, Computer ScienceAdvances on Superelliptic Curves and their Applications
- 2015

In these lectures, the Julia quadratic for any degree $n$ binary form is defined and a survey of a reduction algorithm over $\mathbb Z$ is described based on recent work of Cremona and Stoll.

### On Elliptic Curves and Binary Quartic Forms

- Mathematics
- 2021

. A Dirichlet series is deﬁned whose coeﬃcients are determined by counting certain integral points on the quadratic twists of an elliptic curve. The function deﬁned by this series has a meromorphic…

### Arithmetic matrices for number fields II: Parametrization of rings by binary forms

- Mathematics
- 2018

We show that binary forms of degree n less than seven parameterize rings, generalizing a result of Levi on binary cubic forms parameterizing cubic rings, which can be related to results of Bhargava.…

### Absolute Reduction of Binary Forms

- Mathematics, Computer Science
- 2019

The absolute reduction is introduced and an algorithm to compute the absolutely reduced form of any binary form is given to determine the minimal Weierstrass equation of a superelliptic curve over an integral ring.

### Binary quartic forms with bounded invariants and small Galois groups

- MathematicsPacific Journal of Mathematics
- 2019

In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such…

### A bound on the average rank of j-invariant zero elliptic curves

- Mathematics, Computer Science
- 2013

It is proved that the average rank of j-invariant 0 elliptic curves, when ordered by discriminant, is bounded above by 3 and the number of equivalence classes of these binary quartic forms is counted.

### Some results on binary forms and counting rational points on algebraic varieties

- Mathematics
- 2016

In this thesis we study several problems related to the representation of integers by binary forms and counting rational points on algebraic varieties. In particular, we establish an asymptotic…

### Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

- Mathematics
- 2010

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and…

## References

SHOWING 1-10 OF 34 REFERENCES

### Classical Invariants and 2-descent on Elliptic Curves

- MathematicsJ. Symb. Comput.
- 2001

The results lead to some simplifications to the method first presented in Birch and Swinnerton-Dyer (1963), and can be applied to give a more efficient algorithm for determining Mordell?Weil groups over Q, as well as being more readily extended to other number fields.

### Computing the rank of elliptic curves over real quadratic number fields of class number 1

- Mathematics, Computer ScienceMath. Comput.
- 1999

An algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one is described, extending the one originally described by Birch and Swinnerton-Dyer for curves over Q.

### Algorithms for Modular Elliptic Curves

- Computer Science, Mathematics
- 1992

This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves with remarks on computer implementation and an extensive set of tables giving the results of the author's implementations of the algorithms.

### Computing Cubic Fields in Quasi-Linear Time

- MathematicsANTS
- 1996

An efficient algorithm to generate cubic fields, up to a given discriminant bound, is presented, which the authors hope will prove a useful tool in their computational exploration.

### A course in computational algebraic number theory

- Computer Science, MathematicsGraduate texts in mathematics
- 1993

The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.

### An Introduction to the Algebra of Quantics

- MathematicsNature
- 1895

THE history of the theory of algebraic forms gives a striking example of the fact that the germ of a mathematical doctrine may remain dormant for a long period, and then suddenly develop in a most…

### A fast algorithm to compute cubic fields

- Computer Science, PhysicsMath. Comput.
- 1997

A very fast algorithm to build up tables of cubic fields with discriminant up to 10 11 and complex cubic fields down to -10 11 has been computed.

### An Introduction to the Geometry of Numbers

- Mathematics
- 1959

Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction…

### Theory of algebraic invariants

- Mathematics
- 1993

Preface Introduction Part I. The Elements of Invariant Theory: 1. The forms 2. The linear transformation 3. The concept of an invariant 4. Properties of invariants and covariants 5. The operation…

### Notes on elliptic curves. II.

- Mathematics
- 1963

----------------------------------------------------Nutzungsbedingungen DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung…