Base field 4.4.10512.1
Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -7, 0, 1]))
gp: K = nfinit(Polrev([1, -6, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,-1,1,0]),K([0,1,0,0]),K([-5,-5,0,1]),K([-51,-147,-28,29]),K([-287,-206,63,12])])
gp: E = ellinit([Polrev([-4,-1,1,0]),Polrev([0,1,0,0]),Polrev([-5,-5,0,1]),Polrev([-51,-147,-28,29]),Polrev([-287,-206,63,12])], K);
magma: E := EllipticCurve([K![-4,-1,1,0],K![0,1,0,0],K![-5,-5,0,1],K![-51,-147,-28,29],K![-287,-206,63,12]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-a^2-5a)\) | = | \((a^3-a^2-5a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 4 \) | = | \(4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((128a^3-128a^2-640a)\) | = | \((a^3-a^2-5a)^{15}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1073741824 \) | = | \(4^{15}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{266320696004661}{256} a^{3} + \frac{266320696004661}{256} a^{2} + \frac{1331603480023305}{256} a + \frac{90950209108247}{32} \) | ||
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: | \(1\) |
Generator | $\left(\frac{90202}{64009} a^{3} - \frac{3771}{64009} a^{2} - \frac{662608}{64009} a - \frac{302029}{64009} : \frac{121612401}{16194277} a^{3} - \frac{141522592}{16194277} a^{2} - \frac{702015104}{16194277} a - \frac{57679796}{16194277} : 1\right)$ |
Height | \(1.0758218114330386365153533349269547945\) |
Torsion structure: | \(\Z/3\Z\) |
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: | $\left(-2 a^{3} + a^{2} + 12 a + 11 : 5 a^{3} - 4 a^{2} - 28 a - 8 : 1\right)$ |
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: | \( 1.0758218114330386365153533349269547945 \) | ||
Period: | \( 28.659471545381856813935013773342846714 \) | ||
Tamagawa product: | \( 15 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 2.00481631535149 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-5a)\) | \(4\) | \(15\) | \(I_{15}\) | Split multiplicative | \(-1\) | \(1\) | \(15\) | \(15\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3Cs.1.1 |
\(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
4.1-a
consists of curves linked by isogenies of
degrees dividing 45.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.