Base field \(\Q(\sqrt{2}, \sqrt{5})\)
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 0, -6, 0, 1]))
gp: K = nfinit(Polrev([4, 0, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0]),K([0,0,-1/2,0]),K([0,0,0,0]),K([-89,0,25/2,0]),K([-187,0,29,0])])
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([0,0,-1/2,0]),Polrev([0,0,0,0]),Polrev([-89,0,25/2,0]),Polrev([-187,0,29,0])], K);
magma: E := EllipticCurve([K![0,1,0,0],K![0,0,-1/2,0],K![0,0,0,0],K![-89,0,25/2,0],K![-187,0,29,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^2+6)\) | = | \((1/2a^3-2a)^{2}\cdot(-a^2+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 400 \) | = | \(4^{2}\cdot25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((20a^2-60)\) | = | \((1/2a^3-2a)^{4}\cdot(-a^2+3)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4000000 \) | = | \(4^{4}\cdot25^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{85201943541088}{25} a^{2} - \frac{65088492979664}{25} \) | ||
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 | $\left(\frac{3}{2} a^{3} + 3 a^{2} - 3 a - 7 : -\frac{7}{2} a^{3} - 12 a^{2} - 4 a + 9 : 1\right)$ |
| Height | \(0.84289541153501969768272794349819895677\) |
| 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: | $\left(-\frac{11}{4} a^{2} + 12 : \frac{11}{8} a^{3} - 6 a : 1\right)$ |
| 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: | \( 0.84289541153501969768272794349819895677 \) | ||
| Period: | \( 12.569687648774982149665383871851493932 \) | ||
| Tamagawa product: | \( 9 \) = \(3\cdot3\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.38385970980569 \) | ||
| Analytic order of Ш: | \( 1 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-2a)\) | \(4\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
| \((-a^2+3)\) | \(25\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
400.1-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{5}) \) | 2.2.5.1-80.1-a8 |
| \(\Q(\sqrt{5}) \) | a curve with conductor norm 20480 (not in the database) |