Base field 4.4.18097.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 6 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 6, -7, -1, 1]))
gp: K = nfinit(Polrev([4, 6, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 6, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-3/2,1/2,1/2]),K([-3,5/2,1/2,-1/2]),K([0,1,0,0]),K([-2,-14,6,4]),K([-18,-31,14,13])])
gp: E = ellinit([Polrev([-1,-3/2,1/2,1/2]),Polrev([-3,5/2,1/2,-1/2]),Polrev([0,1,0,0]),Polrev([-2,-14,6,4]),Polrev([-18,-31,14,13])], K);
magma: E := EllipticCurve([K![-1,-3/2,1/2,1/2],K![-3,5/2,1/2,-1/2],K![0,1,0,0],K![-2,-14,6,4],K![-18,-31,14,13]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a-2)\) | = | \((1/2a^3-1/2a^2-5/2a+1)\cdot(a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 12 \) | = | \(3\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-12a^3-7a^2+85a)\) | = | \((1/2a^3-1/2a^2-5/2a+1)^{13}\cdot(a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 6377292 \) | = | \(3^{13}\cdot4\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1049416907}{6377292} a^{3} + \frac{395116529}{2125764} a^{2} - \frac{4876120775}{6377292} a - \frac{743685596}{1594323} \) | ||
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 | $\left(-\frac{1}{2} a^{3} + \frac{7}{2} a^{2} - \frac{7}{2} a - 1 : \frac{5}{2} a^{3} - \frac{43}{2} a^{2} + \frac{31}{2} a + 12 : 1\right)$ |
Height | \(0.018723521992003206299655056793235152724\) |
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: | \( 0.018723521992003206299655056793235152724 \) | ||
Period: | \( 369.26099002554251500994446555103185733 \) | ||
Tamagawa product: | \( 13 \) = \(13\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.67252037755543 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^3-1/2a^2-5/2a+1)\) | \(3\) | \(13\) | \(I_{13}\) | Split multiplicative | \(-1\) | \(1\) | \(13\) | \(13\) |
\((a)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 12.1-d consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.