Base field \(\Q(\zeta_{13})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,2,1,-4,0,1]),K([1,-8,0,6,0,-1]),K([-1,-2,1,1,0,0]),K([-11069,-48724,-7581,36471,3526,-6649]),K([-839159,-3734843,-400010,2939749,227195,-549202])])
gp: E = ellinit([Polrev([-2,2,1,-4,0,1]),Polrev([1,-8,0,6,0,-1]),Polrev([-1,-2,1,1,0,0]),Polrev([-11069,-48724,-7581,36471,3526,-6649]),Polrev([-839159,-3734843,-400010,2939749,227195,-549202])], K);
magma: E := EllipticCurve([K![-2,2,1,-4,0,1],K![1,-8,0,6,0,-1],K![-1,-2,1,1,0,0],K![-11069,-48724,-7581,36471,3526,-6649],K![-839159,-3734843,-400010,2939749,227195,-549202]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1)\) | = | \((1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 1 \) | = | 1 |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((1)\) | = | \((1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1 \) | = | 1 |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -3387888351672962316333 a^{4} + 3387888351672962316333 a^{3} + 13551553406691849265332 a^{2} - 6775776703345924632666 a - 14577323462934449612494 \) | ||
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: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(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: | \( 0.0026716469568848618763422560119573014187 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 0.571393 \) | ||
Analytic order of Ш: | \( 130321 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
No primes of bad reduction.
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 |
\(19\) | 19B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 19 and 57.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 57.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:
Base field | Curve |
---|---|
\(\Q(\sqrt{13}) \) | 2.2.13.1-169.1-a1 |