Base field 3.3.985.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,-1,1]),K([0,0,0]),K([-4,-1,1]),K([-31,-4,4]),K([-66,-8,11])])
gp: E = ellinit([Polrev([-4,-1,1]),Polrev([0,0,0]),Polrev([-4,-1,1]),Polrev([-31,-4,4]),Polrev([-66,-8,11])], K);
magma: E := EllipticCurve([K![-4,-1,1],K![0,0,0],K![-4,-1,1],K![-31,-4,4],K![-66,-8,11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a+3)\) | = | \((-a-1)\cdot(a^2-2a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 55 \) | = | \(5\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((5a^2+4a+24)\) | = | \((-a-1)\cdot(a^2-2a-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 73205 \) | = | \(5\cdot11^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1747148534369}{73205} a^{2} + \frac{3195734287557}{73205} a - \frac{567030464936}{73205} \) | ||
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: | \(1\) |
Generator | $\left(\frac{79}{9} a^{2} - \frac{241}{9} a + \frac{85}{9} : -\frac{2992}{27} a^{2} + \frac{2942}{9} a - 28 : 1\right)$ |
Height | \(4.2825730219944772312732809624325502840\) |
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);
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Torsion generator: | $\left(a^{2} - a - 5 : 2 a^{2} - 2 a - 14 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 4.2825730219944772312732809624325502840 \) | ||
Period: | \( 179.13148461925611399666666449018914566 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
Torsion order: | \(6\) | ||
Leading coefficient: | \( 4.07387160 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a-1)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((a^2-2a-2)\) | \(11\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
55.2-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.