Base field \(\Q(\sqrt{-2}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
gp: K = nfinit(Polrev([2, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,1]),K([468,103]),K([-1770,2360])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,1]),Polrev([468,103]),Polrev([-1770,2360])], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,1],K![468,103],K![-1770,2360]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((102a)\) | = | \((a)^{3}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 20808 \) | = | \(2^{3}\cdot3\cdot3\cdot17\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-205638528a-670270764)\) | = | \((a)^{4}\cdot(-a-1)^{13}\cdot(a-1)\cdot(-2a+3)^{3}\cdot(2a+3)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 533837305469157264 \) | = | \(2^{4}\cdot3^{13}\cdot3\cdot17^{3}\cdot17^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{13596667005814784}{2263710671811} a + \frac{35649488015527936}{2263710671811} \) | ||
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: | \(1\) |
Generator | $\left(-11 a + 6 : -25 a + 10 : 1\right)$ |
Height | \(0.066462902015090049742429994160628057501\) |
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: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.066462902015090049742429994160628057501 \) | ||
Period: | \( 0.40174624818514474706349573972860852027 \) | ||
Tamagawa product: | \( 130 \) = \(2\cdot13\cdot1\cdot1\cdot5\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 4.9089598501982939107355780662682014439 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(2\) | \(III\) | Additive | \(1\) | \(3\) | \(4\) | \(0\) |
\((-a-1)\) | \(3\) | \(13\) | \(I_{13}\) | Split multiplicative | \(-1\) | \(1\) | \(13\) | \(13\) |
\((a-1)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((-2a+3)\) | \(17\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((2a+3)\) | \(17\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
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 20808.5-e consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.