Base field 3.3.229.1
Generator \(a\), with minimal polynomial \( x^{3} - 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -4, 0, 1]))
gp: K = nfinit(Polrev([-1, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0]),K([4,0,-1]),K([-3,1,1]),K([-415,-464,-131]),K([-4350,-7754,-2913])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([4,0,-1]),Polrev([-3,1,1]),Polrev([-415,-464,-131]),Polrev([-4350,-7754,-2913])], K);
magma: E := EllipticCurve([K![1,0,0],K![4,0,-1],K![-3,1,1],K![-415,-464,-131],K![-4350,-7754,-2913]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((a+1)\cdot(a^2-a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 8 \) | = | \(2\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4a^2+12a+12)\) | = | \((a+1)^{2}\cdot(a^2-a-3)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -4096 \) | = | \(-2^{2}\cdot4^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{304458198456664063}{2} a^{2} + \frac{1237813476398146963}{32} a + \frac{19170794207295639649}{32} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(0\) |
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: | $\left(\frac{3}{4} a^{2} - \frac{29}{4} a - \frac{27}{2} : -\frac{7}{8} a^{2} + \frac{25}{8} a + \frac{33}{4} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1.2341965007087875840566161024071329934 \) | ||
Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.0194750053561877888775265541138080627 \) | ||
Analytic order of Ш: | \( 25 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a+1)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((a^2-a-3)\) | \(4\) | \(1\) | \(I_{5}\) | Non-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 |
---|---|
\(2\) | 2B |
\(3\) | 3B |
\(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 5, 6, 10, 15 and 30.
Its isogeny class
8.1-b
consists of curves linked by isogenies of
degrees dividing 30.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.