Base field 5.5.65657.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-6,-8,9,3,-2]),K([2,7,-4,-2,1]),K([1,1,0,0,0]),K([6,-3,-21,-3,5]),K([-4,-33,-40,2,9])])
gp: E = ellinit([Polrev([-6,-8,9,3,-2]),Polrev([2,7,-4,-2,1]),Polrev([1,1,0,0,0]),Polrev([6,-3,-21,-3,5]),Polrev([-4,-33,-40,2,9])], K);
magma: E := EllipticCurve([K![-6,-8,9,3,-2],K![2,7,-4,-2,1],K![1,1,0,0,0],K![6,-3,-21,-3,5],K![-4,-33,-40,2,9]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-2a^3-4a^2+4a+2)\) | = | \((a^4-2a^3-4a^2+4a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 43 \) | = | \(43\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4a^4+5a^3+18a^2-13a-23)\) | = | \((a^4-2a^3-4a^2+4a+2)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -79507 \) | = | \(-43^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{23968185390845}{79507} a^{4} - \frac{32340604970044}{79507} a^{3} + \frac{43993086493009}{79507} a^{2} + \frac{55492835852009}{79507} a + \frac{10232774495362}{79507} \) | ||
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 | $\left(-2 a^{4} + a^{3} + 9 a^{2} - 2 : -2 a^{4} - 3 a^{3} + 11 a^{2} + 16 a : 1\right)$ |
Height | \(0.23469217616037164956641812831120486941\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(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: | \( 0.23469217616037164956641812831120486941 \) | ||
Period: | \( 624.77761162954924657105646691686018884 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.86123519 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4-2a^3-4a^2+4a+2)\) | \(43\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
43.1-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.