Base field 4.4.9248.1
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([2, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([-4,-5,1,1]),K([1,0,0,0]),K([-110,-167,23,37]),K([874,1321,-191,-290])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-4,-5,1,1]),Polrev([1,0,0,0]),Polrev([-110,-167,23,37]),Polrev([874,1321,-191,-290])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-4,-5,1,1],K![1,0,0,0],K![-110,-167,23,37],K![874,1321,-191,-290]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3+a^2-3a)\) | = | \((a)\cdot(a^2+a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 26 \) | = | \(2\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((4a^3-a^2-18a+8)\) | = | \((a)^{2}\cdot(a^2+a-3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -676 \) | = | \(-2^{2}\cdot13^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{305812261}{338} a^{3} - \frac{188599157}{338} a^{2} + \frac{1396429107}{338} a + \frac{918916579}{338} \) | ||
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 | $\left(-\frac{3}{2} a^{3} - a^{2} + 7 a + 5 : \frac{11}{4} a^{3} + \frac{5}{4} a^{2} - \frac{25}{2} a - \frac{17}{2} : 1\right)$ |
Height | \(0.34952776804448572127330846224203641341\) |
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: | $\left(-a^{3} + 3 a + 2 : 5 a^{3} + 6 a^{2} - 26 a - 19 : 1\right)$ |
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: | \( 0.34952776804448572127330846224203641341 \) | ||
Period: | \( 1153.7492885162710904921316193534277351 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(6\) | ||
Leading coefficient: | \( 1.86374590102889 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((a^2+a-3)\) | \(13\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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
26.4-b
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.