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([-4,25]),K([-31,-45])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,1]),Polrev([-4,25]),Polrev([-31,-45])], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,1],K![-4,25],K![-31,-45]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((114a)\) | = | \((a)^{3}\cdot(-a-1)\cdot(a-1)\cdot(-3a+1)\cdot(3a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 25992 \) | = | \(2^{3}\cdot3\cdot3\cdot19\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((24624a+43092)\) | = | \((a)^{4}\cdot(-a-1)^{8}\cdot(a-1)^{4}\cdot(-3a+1)\cdot(3a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 3069603216 \) | = | \(2^{4}\cdot3^{8}\cdot3^{4}\cdot19\cdot19\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{7265024000}{124659} a + \frac{4315648000}{124659} \) | ||
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 | $\left(-2 a + 2 : -2 a - 3 : 1\right)$ |
Height | \(0.063609970384299086359433347999033168151\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.063609970384299086359433347999033168151 \) | ||
Period: | \( 1.7354359690719385699233040404840154882 \) | ||
Tamagawa product: | \( 32 \) = \(2\cdot2^{3}\cdot2\cdot1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 4.9957277642858414010454389126335733428 \) | ||
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 | \(-1\) | \(3\) | \(4\) | \(0\) |
\((-a-1)\) | \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
\((a-1)\) | \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
\((-3a+1)\) | \(19\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((3a+1)\) | \(19\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 25992.5-g 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.