Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,3,-9,-1,2]),K([7,6,-19,-2,4]),K([-2,-3,5,1,-1]),K([-4,-13,9,3,-2]),K([5,13,-1,-3,0])])
gp: E = ellinit([Polrev([4,3,-9,-1,2]),Polrev([7,6,-19,-2,4]),Polrev([-2,-3,5,1,-1]),Polrev([-4,-13,9,3,-2]),Polrev([5,13,-1,-3,0])], K);
magma: E := EllipticCurve([K![4,3,-9,-1,2],K![7,6,-19,-2,4],K![-2,-3,5,1,-1],K![-4,-13,9,3,-2],K![5,13,-1,-3,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3a^4-a^3-14a^2+3a+6)\) | = | \((3a^4-a^3-14a^2+3a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 43 \) | = | \(43\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^4+3a^3+18a^2-9a-11)\) | = | \((3a^4-a^3-14a^2+3a+6)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1849 \) | = | \(-43^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{673276224}{1849} a^{4} + \frac{251558661}{1849} a^{3} + \frac{3275824811}{1849} a^{2} - \frac{548956180}{1849} a - \frac{1829937870}{1849} \) | ||
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 | $\left(-a^{4} + 6 a^{2} - 2 a - 2 : -a^{4} + 3 a^{3} - 4 a - 1 : 1\right)$ |
| Height | \(0.0013956060388066603632085360759871885049\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.0013956060388066603632085360759871885049 \) | ||
| Period: | \( 16404.266367643609770763198008809907217 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.47115856 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3a^4-a^3-14a^2+3a+6)\) | \(43\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 43.1-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.