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([0,0]),K([0,0]),K([-1,-4]),K([0,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-1,-4]),Polrev([0,0])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-1,-4],K![0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-124a-64)\) | = | \((a)^{5}\cdot(-a-1)^{2}\cdot(a+3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 34848 \) | = | \(2^{5}\cdot3^{2}\cdot11^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-7424a-6080)\) | = | \((a)^{12}\cdot(-a-1)^{3}\cdot(a+3)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 147197952 \) | = | \(2^{12}\cdot3^{3}\cdot11^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( 1728 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z[\sqrt{-1}]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(-a - 3 : -3 a + 2 : 1\right)$ |
Height | \(0.27501946578027959628418399480109482150\) |
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);
| |
Torsion generator: | $\left(0 : 0 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.27501946578027959628418399480109482150 \) | ||
Period: | \( 2.8685072740566963672787803493363276369 \) | ||
Tamagawa product: | \( 16 \) = \(2^{2}\cdot2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 4.4626659457239394865029269777626454533 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(4\) | \(I_{3}^{*}\) | Additive | \(1\) | \(5\) | \(12\) | \(0\) |
\((-a-1)\) | \(3\) | \(2\) | \(III\) | Additive | \(1\) | \(2\) | \(3\) | \(0\) |
\((a+3)\) | \(11\) | \(2\) | \(III\) | Additive | \(1\) | \(2\) | \(3\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=2\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
34848.1-h
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.