# Extended Examples ## `Example: ellfunfld1 (ex-4e11bd)` In this example we construct an elliptic curve by using a pencil of cubic curves passing through 8 given points. The 8 points are defined over $GF(1831^8)$ and form a Galois orbit over $GF(1831)$. ```magma > p := 1831; > F := GF(p); > Fe := ext; > K := FunctionField(F); > P2 := ProjectivePlane(K); > // define the 8 points: > points := [ [ u^(p^i), (u^3+u+1)^(p^i) ] : i in [0..7] ]; > M := [ [ p[1]^i*p[2]^j : i in [0..3-j], j in [0..3] ] : p in points ]; > // find the coefficients of 2 cubics that pass through the points: > B := Basis(Kernel(Transpose(Matrix(M)))); > R := PolynomialRing(F, 2); > mono_aff := [ x^i*y^j : i in [0..3-j], j in [0..3] ]; > // f1 and f2 are the cubics: > f1 := &+[ (F!B[1][i])*mono_aff[i] : i in [1..10] ]; > f2 := &+[ (F!B[2][i])*mono_aff[i] : i in [1..10] ]; > // Find the 9th intersection point, which we use as zero, to put > // it to a nice Weierstrass model : > P9 := Points(Scheme(Spec(R), [f1, f2]))[1]; > F1 := Homogenization(Evaluate(f1, [X, Y]), Z); > F2 := Homogenization(Evaluate(f2, [X, Y]), Z); > C := Curve(P2, F1 + t*F2); > E := MinimalDegreeModel(EllipticCurve(C, C![P9[1], P9[2]])); ``` We could transfer the 8 points to this Weierstrass model, and use them to determine the Mordell–Weil group. Instead, we will see what Magma is able to compute just from the Weierstrass model. ```magma > KodairaSymbols(E); [ , , , ] ``` All fibres are irreducible. According to the theory of rational elliptic surfaces, the geometric Mordell–Weil lattice should then be isomorphic to the root lattice $E_8$. We can check this. We also compute the Mordell–Weil lattice over the ground field. ```magma > Lgeom := GeometricMordellWeilLattice(E); > IsIsomorphic(Lgeom, Lattice("E", 8)); true [-1 0 -2 1 1 -1 0 -1] [ 0 0 0 1 0 0 -1 1] [ 2 0 2 -2 -1 1 1 0] [-2 0 -1 0 1 0 0 -1] [ 1 0 0 1 -1 -1 -1 1] [-1 0 0 0 1 0 1 -1] [ 2 -1 2 -3 -1 1 1 1] [ 1 -1 1 -1 0 0 0 0] > L, f := MordellWeilLattice(E); > L; Standard Lattice of rank 1 and degree 1 Inner Product Matrix: [8] > f(L.1); ((1057*t^8 + 384*t^7 + 351*t^6 + 728*t^5 + 872*t^4 + 948*t^3 + 1473*t^2 + 257*t + 1333)/(t^6 + 100*t^5 + 1565*t^4 + 1145*t^3 + 927*t^2 + 1302*t + 1197) : (1202*t^12 + 1506*t^11 + 1718*t^10 + 1365*t^9 + 656*t^8 + 325*t^7 + 1173*t^6 + 902*t^5 + 1555*t^4 + 978*t^3 + 616*t^2 + 779*t + 964)/(t^9 + 150*t^8 + 1520*t^7 + 747*t^6 + 1065*t^5 + 340*t^4 + 1618*t^3 + 1669*t^2 + 1150*t + 1768) : 1) ``` That the rank equals 1 is not surprising; this point corresponds to the ${\mathbb{F}}_{1831}$-rational degree 8 divisor consisting of the sum of the 8 points we started with. To determine the $L$-function of $E$ one would normally have to count points over various extension fields of ${\mathbb{F}}_{1831}$, in this case up to ${\mathbb{F}}_{1831^4}$. This would be costly using current techniques. But since Magma is able to determine the geometric Mordell–Weil lattice, it can compute the $L$-function by simply considering the Galois action on points. Magma automatically uses this when asked for the $L$-function: ```magma > LFunction(E); -126330075128099803866555841*T^8 + 1 ``` ## `Example: Reductionmodp (ex-198933)` In this example we determine the ${\mathbb{C}}(t)$-rank of an elliptic curve, following [[Kloosterman, 2007](../../references.md#cite-kloosterman)]. ```magma > K := FunctionField(Rationals()); > E := EllipticCurve([-(2*t-1)^3*(4*t-1)^2, t*(2*t-1)^3*(4*t-1)^3]); ``` To determine where the surface has bad reduction we determine the primes at which singular fibres collapse. ```magma > &*BadPlaces(E); (t^5 - 99/32*t^4 + 337/128*t^3 - 251/256*t^2 + 3/16*t - 1/64)/t > Discriminant(Numerator($1)); -87887055375/4503599627370496 > Factorisation(Numerator($1)); [ <3, 15>, <5, 3>, <7, 2> ] > Factorisation(Denominator($2)); [ <2, 52> ] ``` So 11 and 13 are the smallest primes of good reduction. We remark that in [[Kloosterman, 2007](../../references.md#cite-kloosterman)] the primes 17 and 19 were used. ```magma > K11 := FunctionField(GF(11)); > E11 := ChangeRing(E,K11); // Reduce E mod 11 > LFunction(E11); 161051*T^5 - 7986*T^4 - 363*T^3 - 33*T^2 - 6*T + 1 > AnalyticInformation(E11); <0, 1, 1> > AnalyticInformation(E11, 2); <1, 1, 35/2> ``` So modulo 11 the rank is 0, but over ${\mathbb{F}}_{11^2}(t)$ the rank equals the geometric rank of 1, and the height of a generator is congruent to ${35 \over 2}$ modulo ${\mathbb{Q}}^2$. From this it can be concluded that the geometric rank in characteristic 0 is at most 1. As the $L$-function has odd degree 5 for any $p$ of good reduction, it will always have a zero at $1/p$ or $-1/p$ and consequently the geometric rank modulo $p$ will always be at least 1. To determine what the geometric rank is in characteristic zero, one can combine information at 2 different primes. ```magma > K13 := FunctionField(GF(13)); > E13 := ChangeRing(E, K13); // Redude E mod 13 > AnalyticInformation(E13); <0, 1, 1> > AnalyticInformation(E13,2); <1, 1, 121/2> ``` So over ${\mathbb{F}}_{13^2}(t)$ the rank equals the geometric rank of 1. The height of a generator is congruent to ${121 \over 2}$ modulo ${\mathbb{Q}}^2$. As the quotient of the heights of generators in characteristics 11 and 13 is not a square in ${\mathbb{Q}}$, there cannot exist a Mordell–Weil group in characteristic zero that both modulo 11 and 13 reduces to a finite index subgroup of the Mordell–Weil group modulo $p$. Hence one can conclude that the geometric Mordell–Weil rank in characteristic zero is zero. ## `Example: L Functionbyhand (ex-98b1c7)` In this example we calculate part of the $L$-function of an elliptic curve for which it is not feasible to compute the $L$-function completely. The simplest way to compute an $L$-function is to call `LFunction`, which counts points on the surface over certain constant field extensions. However, if the required extension fields are too big then `LFunction` will not terminate. One can determine the required extension degrees as follows. ```magma > K := FunctionField(GF(5)); > E := EllipticCurve([t^9+t^2, t^14+t^4+t^3]); > h2 := BettiNumber(E, 2); > N := FrobeniusActionOnTrivialLattice(E); > [h2, h2 - NumberOfRows(N)]; [ 34, 21 ] ``` So the $H^2$ has dimension 34, of which a 13-dimensional piece is generated by the trivial lattice and a 21-dimensional piece is unknown. To determine the $L$-function one would have to at least count points over ${\mathbb{F}}_{5^{10}}$. As this is not feasible we will instead count points over extension degrees up to 5 and print the first few coefficients of the $L$-function. ```magma > nop := NumbersOfPointsOnSurface(E, 5); > traces := [ nop[i] - 1 - 25^i - 5^i*Trace(N^i) : i in [1..5] ]; > CharacteristicPolynomialFromTraces(traces); 5750*T^5 + 875*T^4 - 40*T^3 - 40*T^2 + 1 ```