Base field 3.3.361.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, -6, -1, 1]))
gp: K = nfinit(Polrev([7, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0]),K([4,-1,-1]),K([-4,0,1]),K([12,-1,-2]),K([5,0,-1])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([4,-1,-1]),Polrev([-4,0,1]),Polrev([12,-1,-2]),Polrev([5,0,-1])], K);
magma: E := EllipticCurve([K![1,1,0],K![4,-1,-1],K![-4,0,1],K![12,-1,-2],K![5,0,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((3)\) | = | \((3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27 \) | = | \(27\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-3)\) | = | \((3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -27 \) | = | \(-27\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{2375}{3} \) | ||
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 | $\left(\frac{12849}{2401} a^{2} + \frac{17928}{2401} a - \frac{5179}{343} : -\frac{5027331}{117649} a^{2} - \frac{6269846}{117649} a + \frac{2247488}{16807} : 1\right)$ |
Height | \(3.5712293443622126639241266809074098791\) |
Torsion structure: | \(\Z/7\Z\) |
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: | $\left(0 : -1 : 1\right)$ |
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: | \( 3.5712293443622126639241266809074098791 \) | ||
Period: | \( 155.64397066533495533947782968372936928 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(7\) | ||
Leading coefficient: | \( 1.7911073533182458002224241964941132866 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3)\) | \(27\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(7\) | 7B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
27.1-b
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:
Base field | Curve |
---|---|
\(\Q\) | 1083.c2 |