Base field 3.3.148.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -1, 1]))
gp: K = nfinit(Polrev([1, -3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0]),K([-3,-2,1]),K([1,0,0]),K([173905,-441567,-377381]),K([106974682,-271628927,-232144315])])
gp: E = ellinit([Polrev([0,0,0]),Polrev([-3,-2,1]),Polrev([1,0,0]),Polrev([173905,-441567,-377381]),Polrev([106974682,-271628927,-232144315])], K);
magma: E := EllipticCurve([K![0,0,0],K![-3,-2,1],K![1,0,0],K![173905,-441567,-377381],K![106974682,-271628927,-232144315]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^2+3a+4)\) | = | \((a^2-a-1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^2+a+3)\) | = | \((a^2-a-1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 125 \) | = | \(5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( 491380736 a^{2} + 574509056 a - 227098624 \) | ||
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: | $\left(-\frac{273}{2} a^{2} - \frac{317}{2} a + 64 : -\frac{1}{2} : 1\right)$ |
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: | \( 10.860736231663658624898432923697025049 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 0.44637350946902694342944494420356594233 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-a-1)\) | \(5\) | \(2\) | \(III\) | Additive | \(-1\) | \(2\) | \(3\) | \(0\) |
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 |
\(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
25.2-b
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.