Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-3,-1,1]),K([0,-5,-1,1]),K([1,1,0,0]),K([-1404,2629,506,-544]),K([-33259,61674,14266,-13092])])
gp: E = ellinit([Polrev([2,-3,-1,1]),Polrev([0,-5,-1,1]),Polrev([1,1,0,0]),Polrev([-1404,2629,506,-544]),Polrev([-33259,61674,14266,-13092])], K);
magma: E := EllipticCurve([K![2,-3,-1,1],K![0,-5,-1,1],K![1,1,0,0],K![-1404,2629,506,-544],K![-33259,61674,14266,-13092]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a-3)\) | = | \((-a^3+a^2+4a)\cdot(a^3-a^2-4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-39a^3+24a^2+152a-39)\) | = | \((-a^3+a^2+4a)\cdot(a^3-a^2-4a+1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 17294403 \) | = | \(3\cdot7^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{73204218707003465754418530011323778}{17294403} a^{3} - \frac{29465194712745950920219395499999665}{5764801} a^{2} - \frac{110828615043320066217968092545904435}{5764801} a + \frac{60623589840441817561364743725976875}{5764801} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-\frac{93774}{3025} a^{3} + \frac{277796}{3025} a^{2} + \frac{35072}{605} a - \frac{117993}{3025} : -\frac{160275702}{166375} a^{3} + \frac{358923458}{166375} a^{2} + \frac{77510956}{33275} a - \frac{272626139}{166375} : 1\right)$ |
| Height | \(6.6182378136884860742154484345434523973\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{21}{4} a^{3} + \frac{5}{2} a^{2} + \frac{103}{4} a - \frac{57}{4} : \frac{109}{8} a^{3} - 13 a^{2} - \frac{189}{4} a + 25 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 6.6182378136884860742154484345434523973 \) | ||
| Period: | \( 1.3802803304060593042144664110623292779 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.93659957703088 \) | ||
| Analytic order of Ш: | \( 16 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((a^3-a^2-4a+1)\) | \(7\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
21.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.