Base field 3.3.733.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
gp: K = nfinit(Polrev([8, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,0,1]),K([4,0,-1]),K([0,1,0]),K([1017,-482,-318]),K([9014,-4306,-2837])])
gp: E = ellinit([Polrev([-4,0,1]),Polrev([4,0,-1]),Polrev([0,1,0]),Polrev([1017,-482,-318]),Polrev([9014,-4306,-2837])], K);
magma: E := EllipticCurve([K![-4,0,1],K![4,0,-1],K![0,1,0],K![1017,-482,-318],K![9014,-4306,-2837]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2+4a-4)\) | = | \((a^2-6)^{2}\cdot(a^2+2a-3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 196 \) | = | \(2^{2}\cdot7^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-40a^2+167a+56)\) | = | \((a^2-6)^{8}\cdot(a^2+2a-3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -30118144 \) | = | \(-2^{8}\cdot7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -12237 a^{2} - 1963 a + 84888 \) | ||
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 | $\left(9 a^{2} + 13 a - 29 : -47 a^{2} - 72 a + 148 : 1\right)$ |
Height | \(0.12500604457827029485247055169952963781\) |
Torsion structure: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-a^{2} - 2 a + 2 : 2 a^{2} + 2 a - 8 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.12500604457827029485247055169952963781 \) | ||
Period: | \( 81.375290865936847260349038100372080809 \) | ||
Tamagawa product: | \( 6 \) = \(3\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.6907682890244398666539308702642294383 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-6)\) | \(2\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
\((a^2+2a-3)\) | \(7\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
196.4-d
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.