Base field \(\Q(\sqrt{5}) \)
Generator \(\phi\), with minimal polynomial \( x^{2} - x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -1, 1]))
gp: K = nfinit(Polrev([-1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([1,-1]),K([1,0]),K([231,-240]),K([-19762,11763])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,-1]),Polrev([1,0]),Polrev([231,-240]),Polrev([-19762,11763])], K);
magma: E := EllipticCurve([K![0,0],K![1,-1],K![1,0],K![231,-240],K![-19762,11763]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((57)\) | = | \((3)\cdot(4\phi-3)\cdot(-4\phi+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 3249 \) | = | \(9\cdot19\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-19072827\phi-396041643)\) | = | \((3)^{10}\cdot(4\phi-3)^{5}\cdot(-4\phi+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 164038844002102281 \) | = | \(9^{10}\cdot19^{5}\cdot19\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{55744326282944512}{146211169851} \phi - \frac{34449182475366400}{146211169851} \) | ||
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 | $\left(-\frac{549}{20} \phi + \frac{663}{10} : -\frac{9477}{25} \phi + \frac{107063}{200} : 1\right)$ |
Height | \(4.2536673280398014220853340876339238486\) |
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);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 4.2536673280398014220853340876339238486 \) | ||
Period: | \( 0.31086053256673059080344558631384123277 \) | ||
Tamagawa product: | \( 2 \) = \(2\cdot1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.3653973032334236324114867836122771811 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3)\) | \(9\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
\((4\phi-3)\) | \(19\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
\((-4\phi+1)\) | \(19\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(5\) | 5B.1.4[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
3249.1-e
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.