Base field 4.4.13068.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-6,-1,1]),K([-4,-3,1,0]),K([-1,5,2,-1]),K([56,-84,-279,98]),K([578,-895,-2980,1058])])
gp: E = ellinit([Polrev([-1,-6,-1,1]),Polrev([-4,-3,1,0]),Polrev([-1,5,2,-1]),Polrev([56,-84,-279,98]),Polrev([578,-895,-2980,1058])], K);
magma: E := EllipticCurve([K![-1,-6,-1,1],K![-4,-3,1,0],K![-1,5,2,-1],K![56,-84,-279,98],K![578,-895,-2980,1058]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-a^2-5a-4)\) | = | \((2a^3-3a^2-10a+2)\cdot(a^2+a-1)\) |
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: | \((-3a^3+9a^2+3a-6)\) | = | \((2a^3-3a^2-10a+2)^{4}\cdot(a^2+a-1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1296 \) | = | \(3^{4}\cdot4^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{7097336741951}{3} a^{3} + \frac{58010293511921}{12} a^{2} + 520639444971 a - \frac{9321083623331}{12} \) | ||
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: | \(0\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(\frac{9}{4} a^{3} - \frac{25}{4} a^{2} - \frac{3}{4} a + 1 : \frac{5}{8} a^{2} - \frac{21}{8} a + \frac{5}{8} : 1\right)$ | $\left(a^{3} - 3 a^{2} - 2 a + 1 : -a : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 558.18371671463779601434342641888060278 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 2.44142059926107 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2a^3-3a^2-10a+2)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((a^2+a-1)\) | \(4\) | \(2\) | \(I_{2}\) | 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 |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
12.1-e
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.