Base field 4.4.11324.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 4 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([2, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-3,0,1]),K([-2,3,1,-1]),K([1,-4,0,1]),K([-1,0,1,-1]),K([0,5,-2,-3])])
gp: E = ellinit([Polrev([1,-3,0,1]),Polrev([-2,3,1,-1]),Polrev([1,-4,0,1]),Polrev([-1,0,1,-1]),Polrev([0,5,-2,-3])], K);
magma: E := EllipticCurve([K![1,-3,0,1],K![-2,3,1,-1],K![1,-4,0,1],K![-1,0,1,-1],K![0,5,-2,-3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-3a)\) | = | \((a^3-5a)\cdot(-a^2+a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(2\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^2+4a-4)\) | = | \((a^3-5a)^{8}\cdot(-a^2+a+3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1024 \) | = | \(2^{8}\cdot4\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{174483}{16} a^{3} - 234 a^{2} + \frac{466917}{8} a + \frac{352373}{16} \) | ||
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 | $\left(a + 1 : -3 a^{3} - 2 a^{2} + 10 a + 2 : 1\right)$ |
| Height | \(0.37445390025906451375872799079735884855\) |
| 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: | $\left(-\frac{3}{4} a^{2} - \frac{1}{2} a + \frac{1}{2} : \frac{5}{8} a^{3} + \frac{5}{4} a^{2} - \frac{1}{4} a - 2 : 1\right)$ |
| 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: | \( 0.37445390025906451375872799079735884855 \) | ||
| Period: | \( 266.03421493341625030210167234789886618 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.87225896623124 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-5a)\) | \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((-a^2+a+3)\) | \(4\) | \(1\) | \(I_{1}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
8.1-c
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.