Base field 5.5.65657.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0,0]),K([3,7,-4,-2,1]),K([-4,-7,5,2,-1]),K([-314,-1577,-1529,154,330]),K([-6558,-38029,-41530,3425,9329])])
gp: E = ellinit([Polrev([0,1,0,0,0]),Polrev([3,7,-4,-2,1]),Polrev([-4,-7,5,2,-1]),Polrev([-314,-1577,-1529,154,330]),Polrev([-6558,-38029,-41530,3425,9329])], K);
magma: E := EllipticCurve([K![0,1,0,0,0],K![3,7,-4,-2,1],K![-4,-7,5,2,-1],K![-314,-1577,-1529,154,330],K![-6558,-38029,-41530,3425,9329]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-4a)\) | = | \((-a^4+a^3+4a^2-2a-2)^{2}\cdot(-a^2+a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 45 \) | = | \(3^{2}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2a^4+5a^3+7a^2-12a-7)\) | = | \((-a^4+a^3+4a^2-2a-2)^{6}\cdot(-a^2+a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -3645 \) | = | \(-3^{6}\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \) | ||
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: | \(\Z/3\Z\) |
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: | $\left(-11 a^{4} - a^{3} + 49 a^{2} + 34 a + 8 : 83 a^{4} + 42 a^{3} - 378 a^{2} - 394 a - 70 : 1\right)$ |
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: | \( 134.14023065512017269522700963416277868 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 1.45417284 \) | ||
Analytic order of Ш: | \( 25 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
\((-a^2+a+2)\) | \(5\) | \(1\) | \(I_{1}\) | 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 |
---|---|
\(3\) | 3B.1.1 |
\(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
45.1-b
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.