Base field 4.4.1957.1
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -4, 0, 1]))
gp: K = nfinit(Polrev([1, -1, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0,0]),K([-1,-4,0,1]),K([-1,3,1,-1]),K([-177,-257,29,69]),K([-1004,-1515,140,387])])
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([-1,-4,0,1]),Polrev([-1,3,1,-1]),Polrev([-177,-257,29,69]),Polrev([-1004,-1515,140,387])], K);
magma: E := EllipticCurve([K![0,0,0,0],K![-1,-4,0,1],K![-1,3,1,-1],K![-177,-257,29,69],K![-1004,-1515,140,387]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-5a)\) | = | \((a^3-5a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((6a^3+51a^2+6a-68)\) | = | \((a^3-5a)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -28629151 \) | = | \(-31^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{180825454881646489600000}{28629151} a^{3} + \frac{318978801202372556800000}{28629151} a^{2} + \frac{160618453429727667920896}{28629151} a - \frac{102507894005169170706432}{28629151} \) | ||
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 | $\left(-\frac{29}{9} a^{3} - \frac{17}{9} a^{2} + \frac{52}{3} a + \frac{149}{9} : \frac{212}{27} a^{3} - \frac{16}{27} a^{2} - \frac{412}{9} a - \frac{998}{27} : 1\right)$ |
| Height | \(1.0227612151799583536850781090975827728\) |
| 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: | \( 1.0227612151799583536850781090975827728 \) | ||
| Period: | \( 2.3999846553071537014353139279692778687 \) | ||
| Tamagawa product: | \( 5 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.10972992737054 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-5a)\) | \(31\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
31.1-a
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.