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([1,0]),K([-1,-1]),K([1,1]),K([-57,-30]),K([-131,-101])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,1]),Polrev([-57,-30]),Polrev([-131,-101])], K);
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,1],K![-57,-30],K![-131,-101]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((51a)\) | = | \((a)\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 5202 \) | = | \(2\cdot3\cdot3\cdot17\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((894336a-1036320)\) | = | \((a)^{10}\cdot(-a-1)\cdot(a-1)^{11}\cdot(-2a+3)^{2}\cdot(2a+3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2673632904192 \) | = | \(2^{10}\cdot3\cdot3^{11}\cdot17^{2}\cdot17\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{2091127194353}{409563864} a + \frac{36587816460805}{1638255456} \) | ||
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: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-a - 5 : 2 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1.0907955024834311850447312639986464508 \) | ||
Tamagawa product: | \( 20 \) = \(( 2 \cdot 5 )\cdot1\cdot1\cdot2\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.8565444834691086585392721553521688807 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
\((-a-1)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((a-1)\) | \(3\) | \(1\) | \(I_{11}\) | Non-split multiplicative | \(1\) | \(1\) | \(11\) | \(11\) |
\((-2a+3)\) | \(17\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((2a+3)\) | \(17\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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.
Its isogeny class
5202.5-h
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.