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([-2,0,1]),K([4,-1,-1]),K([0,1,0]),K([214,746,-403]),K([-4017,-13910,7473])])
gp: E = ellinit([Polrev([-2,0,1]),Polrev([4,-1,-1]),Polrev([0,1,0]),Polrev([214,746,-403]),Polrev([-4017,-13910,7473])], K);
magma: E := EllipticCurve([K![-2,0,1],K![4,-1,-1],K![0,1,0],K![214,746,-403],K![-4017,-13910,7473]]);
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: | \((-70a^2+50a+98)\) | = | \((a+1)\cdot(a^2-a-3)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2097152 \) | = | \(2\cdot4^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{410180291815113}{1024} a^{2} - \frac{381468430679405}{512} a - \frac{221127400043935}{1024} \) | ||
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: | \(\Z/6\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(6 a^{2} - 3 a - 1 : -10 a^{2} - 27 a - 6 : 1\right)$ |
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: | \( 36.531488933205926888813769601209846752 \) | ||
Tamagawa product: | \( 10 \) = \(1\cdot( 2 \cdot 5 )\) | ||
Torsion order: | \(6\) | ||
Leading coefficient: | \( 0.67057464970504837012570298903524365768 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a+1)\) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((a^2-a-3)\) | \(4\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
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.1.1 |
\(5\) | 5B.4.2 |
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-a
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.