Base field 4.4.2624.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -3, -2, 1]))
gp: K = nfinit(Polrev([1, 2, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-4,-1,1]),K([1,1,-3,1]),K([0,-3,-1,1]),K([-58,-205,-72,50]),K([-414,-1331,-425,314])])
gp: E = ellinit([Polrev([1,-4,-1,1]),Polrev([1,1,-3,1]),Polrev([0,-3,-1,1]),Polrev([-58,-205,-72,50]),Polrev([-414,-1331,-425,314])], K);
magma: E := EllipticCurve([K![1,-4,-1,1],K![1,1,-3,1],K![0,-3,-1,1],K![-58,-205,-72,50],K![-414,-1331,-425,314]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^3-4a^2-4a+3)\) | = | \((2a^3-4a^2-4a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2a^3+4a^2+4a-3)\) | = | \((2a^3-4a^2-4a+3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 49 \) | = | \(49\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{2220003697148358852}{7} a^{3} + \frac{7166898260661020127}{7} a^{2} - \frac{2143289518391363318}{7} a - 258190746641008395 \) | ||
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 | $\left(-\frac{1}{49} a^{3} + \frac{46}{7} a^{2} - \frac{578}{49} a - \frac{213}{49} : -\frac{2832}{343} a^{3} + \frac{1017}{49} a^{2} - \frac{4069}{343} a - \frac{3358}{343} : 1\right)$ |
| Height | \(2.2247839230664694882723028092129402093\) |
| 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: | $\left(-3 a^{3} + 4 a^{2} + 14 a + 5 : -6 a^{3} + 6 a^{2} + 20 a + 5 : 1\right)$ |
| 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: | \( 2.2247839230664694882723028092129402093 \) | ||
| Period: | \( 4.1955585659374044447679625985728499076 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.63997874508118 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^3-4a^2-4a+3)\) | \(49\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 and 6.
Its isogeny class
49.2-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.