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In [[mathematics]], especially in [[algebraic geometry]], a '''quartic surface''' is a [[surface (mathematics)|surface]] defined by an equation of ...ly related types of quartic surface: affine and projective. An ''affine'' quartic surface is the solution set of an equation of the form ...

...''CR''₄ or the '''Castelnuovo–Richmond–Igusa quartic''') is a quartic [[hypersurface]] in 4-dimensional [[projective space]], studied by {{harvs| It is closely related to the [[moduli space]] of genus 2 curves with level 2 structure. It is the dual of the [[Segre cubic]]. ...

...nal operations. The rows of the table are for different models of elliptic curves. These are the operations considered: ...ter the table. For information about other possible operations on elliptic curves see http://hyperelliptic.org/EFD/g1p/index.html. ...

{{Short description|Mathematical curves that are isomorphic over algebraic closures}} ...e isogeny. Some curves have higher order twists such as '''cubic''' and '''quartic twists'''. The curve and its twists have the same [[j-invariant]]. ...

...rch and Swinnerton-Dyer conjecture]] for a positive proportion of elliptic curves. ...

{{short description|Class of geometric plane curves}} ...on]] of distances from two fixed points ([[Focus (geometry)|foci]]). These curves are named after French mathematician [[René Descartes]], who used them in [ ...

{{Short description|Type of quartic plane curve}} ...''3-petaled rose curve''', and '''paquerette de mélibée''') is a type of [[quartic plane curve]]. The name comes from the Latin terms for 3-leaved, defining i ...

...on''', that is given by an intersection of two surfaces, and the '''Jacobi quartic'''. ==Elliptic Curves: Basics== ...

...rves. The basic idea would be to use higher degree equations. Consider the quartic equation ...ion formula]]; for details, see {{slink|Adjunction formula|Applications to curves}}.<ref>''Algebraic geometry'', [[Robin Hartshorne]], Springer GTM 52, {{ISB ...

...ic]], '''the Bring sextic''') is the [[Algebraic curve#Non-plane algebraic curves|curve]] in the [[projective space]] <math>\mathbb{P}^4</math> cut out by th Similarly to the [[Klein quartic]], Bring's surface does not maximize the systole length among compact Riema ...

{{short description|Quartic plane curve}} ...[limaçon]]. The shape of the limaçon trisectrix can be specified by other curves particularly as a [[rose (mathematics)|rose]], [[conchoid (mathematics)|con ...

{{Short description|Curves in algebraic geometric}} ...ic curves are important in [[public key cryptography]] and twisted Edwards curves are at the heart of an electronic signature scheme called [[EdDSA]] that of ...

| title = Construction of class fields and zeta functions of algebraic curves | contribution = The Klein quartic in number theory ...

...of cuttings along non-intersecting [[Curve#Topological_curve|closed simple curves]] without rendering the resultant [[manifold]] disconnected.<ref>Munkres, J ...om a property of [[Weierstrass's elliptic functions]] that allows elliptic curves to be obtained from the quotient of the [[complex plane]] by a [[fundamenta ...

...\mathbb{Q}</math> to be bounded or not. It has been shown that there exist curves with rank at least 29,<ref name="largest"> ...have rank 1; all remaining ranks consist of a total of 0% of all elliptic curves over <math>\mathbb{Q}</math>. ...

...torsion conjecture has been completely resolved in the case of [[elliptic curves]]. ==Elliptic curves== ...

...sed field of characteristic zero is ruled. Smooth cubic 3-folds and smooth quartic 3-folds in '''P'''⁴ over '''C''' are uniruled but not ruled. ...=Bogomolov | first1=Fedor | last2=Tschinkel | first2=Yuri | title=Rational curves and points on K3 surfaces | journal=American Journal of Mathematics | volum ...

[[File:Peano-flaeche-12.svg|thumb|upright=1.2|Peano surface and its level curves for level 0 (parabolas, green and purple)]] The Peano surface is a [[quartic surface]]. ...

{{Short description|Family of elliptic curves used in cryptography}} [[File:Edward-curves.svg|right|thumb|Edwards curves of equation ''x''²&nbsp;+&nbsp;''y''²&nbsp;=&nbsp;1&n ...

...es=Ph.D. thesis|publisher=University of Exeter|year=1995|title=Descents on Curves of Genus I|pages=16–17}}</ref> The [[elliptic curve]] describing the square | title = Crossed ladders and Euler's quartic ...