Base field \(\Q(\sqrt{13}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, -1, 1]))
gp: K = nfinit(Polrev([-3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([0,1]),K([-24975,10703]),K([-2043804,886340])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,1]),Polrev([-24975,10703]),Polrev([-2043804,886340])], K);
magma: E := EllipticCurve([K![1,1],K![0,0],K![0,1],K![-24975,10703],K![-2043804,886340]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((42)\) | = | \((-a)\cdot(-a+1)\cdot(2)\cdot(7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 1764 \) | = | \(3\cdot3\cdot4\cdot49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-497757560832)\) | = | \((-a)^{11}\cdot(-a+1)^{11}\cdot(2)^{13}\cdot(7)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 247762589365422180532224 \) | = | \(3^{11}\cdot3^{11}\cdot4^{13}\cdot49^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{5022437771811277}{497757560832} \) | ||
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: | \( 0.094334852119462218817186577627962933444 \) | ||
Tamagawa product: | \( 121 \) = \(11\cdot11\cdot1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.1658174393833916787206341599810855087 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(3\) | \(11\) | \(I_{11}\) | Split multiplicative | \(-1\) | \(1\) | \(11\) | \(11\) |
\((-a+1)\) | \(3\) | \(11\) | \(I_{11}\) | Split multiplicative | \(-1\) | \(1\) | \(11\) | \(11\) |
\((2)\) | \(4\) | \(1\) | \(I_{13}\) | Non-split multiplicative | \(1\) | \(1\) | \(13\) | \(13\) |
\((7)\) | \(49\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 1764.1-a consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.