Base field 4.4.16997.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - x + 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, -1, -6, 0, 1]))
gp: K = nfinit(Polrev([5, -1, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -1, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-4,-1,1]),K([3,-5,-1,1]),K([2,-4,-1,1]),K([398,316,-68,-59]),K([-4142,-3904,650,750])])
gp: E = ellinit([Polrev([3,-4,-1,1]),Polrev([3,-5,-1,1]),Polrev([2,-4,-1,1]),Polrev([398,316,-68,-59]),Polrev([-4142,-3904,650,750])], K);
magma: E := EllipticCurve([K![3,-4,-1,1],K![3,-5,-1,1],K![2,-4,-1,1],K![398,316,-68,-59],K![-4142,-3904,650,750]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^2+2)\) | = | \((-a^2+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 7 \) | = | \(7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-15a^3+46a^2+108a-229)\) | = | \((-a^2+2)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -282475249 \) | = | \(-7^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{500718065734107078}{282475249} a^{3} + \frac{1176400322528301747}{282475249} a^{2} - \frac{240441686104387112}{282475249} a - \frac{1065616788500768418}{282475249} \) | ||
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: | $\left(-\frac{7}{4} a^{3} - \frac{1}{4} a^{2} + \frac{35}{4} a + \frac{7}{2} : \frac{5}{8} a^{3} - \frac{3}{2} a^{2} - \frac{25}{8} a + \frac{15}{2} : 1\right)$ |
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: | \( 78.139627726478289970501309619317214500 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.19871294761170 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^2+2)\) | \(7\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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
7.1-a
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.