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([-1,0,1]),K([-1,0,0]),K([-2,-1,1]),K([-795941171665179475905777,2021044966504285468600887,1727261180779898112178031]),K([-128973872132688546804017272390232511682,327489021002657174715208924802606509852,279884457042981407569057081184688558284])])
gp: E = ellinit([Polrev([-1,0,1]),Polrev([-1,0,0]),Polrev([-2,-1,1]),Polrev([-795941171665179475905777,2021044966504285468600887,1727261180779898112178031]),Polrev([-128973872132688546804017272390232511682,327489021002657174715208924802606509852,279884457042981407569057081184688558284])], K);
magma: E := EllipticCurve([K![-1,0,1],K![-1,0,0],K![-2,-1,1],K![-795941171665179475905777,2021044966504285468600887,1727261180779898112178031],K![-128973872132688546804017272390232511682,327489021002657174715208924802606509852,279884457042981407569057081184688558284]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((a^2-a-2)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 8 \) | = | \(2^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2a+2)\) | = | \((a^2-a-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -2624 a^{2} + 1536 a + 9024 \) | ||
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: | \(\Z/8\Z\) |
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: | $\left(6677273311789 a^{2} + 7812987269633 a - 3076961841355 : 45536113168921811620 a^{2} + 53281190672429189413 a - 20983547637181775978 : 1\right)$ |
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: | \( 223.18826420022044179173853209997750472 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 0.57331132207744951681123408265720299983 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-a-2)\) | \(2\) | \(2\) | \(III\) | Additive | \(-1\) | \(3\) | \(4\) | \(0\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
8.1-a
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.