Base field 4.4.13968.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 8, -7, -2, 1]))
gp: K = nfinit(Polrev([4, 8, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 8, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-2,-1/2,1/2]),K([-1,0,0,0]),K([1,1,0,0]),K([-19,-29/2,3/2,2]),K([-20,-213/2,-10,29/2])])
gp: E = ellinit([Polrev([1,-2,-1/2,1/2]),Polrev([-1,0,0,0]),Polrev([1,1,0,0]),Polrev([-19,-29/2,3/2,2]),Polrev([-20,-213/2,-10,29/2])], K);
magma: E := EllipticCurve([K![1,-2,-1/2,1/2],K![-1,0,0,0],K![1,1,0,0],K![-19,-29/2,3/2,2],K![-20,-213/2,-10,29/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-1/2a^3+a^2+5/2a-1)\) | = | \((1/2a^2+1/2a-1)\cdot(-1/2a^3+1/2a^2+4a-1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 8 \) | = | \(2\cdot2^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^3-7a-2)\) | = | \((1/2a^2+1/2a-1)^{3}\cdot(-1/2a^3+1/2a^2+4a-1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -128 \) | = | \(-2^{3}\cdot2^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1552959}{8} a^{3} + 1299159 a^{2} - \frac{21733839}{8} a + \frac{7144713}{4} \) | ||
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: | \(\Z/2\Z\) |
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: | $\left(-\frac{1}{2} a^{3} - \frac{1}{4} a^{2} + 4 a + \frac{15}{4} : -\frac{1}{4} a^{3} + \frac{7}{8} a^{2} + 3 a - \frac{9}{8} : 1\right)$ |
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: | \( 414.78224061954875346435220388596348985 \) | ||
Tamagawa product: | \( 3 \) = \(3\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.63217223361618 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^2+1/2a-1)\) | \(2\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
\((-1/2a^3+1/2a^2+4a-1)\) | \(2\) | \(1\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
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 |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
8.2-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.