Base field \(\Q(\sqrt{2}, \sqrt{5})\)
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 0, -6, 0, 1]))
gp: K = nfinit(Polrev([4, 0, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1,1/2,1/2]),K([1,1,-1/2,-1/2]),K([-1,-1,1/2,1/2]),K([22,243,-248,-153]),K([-2955,-2876,10239/2,2540])])
gp: E = ellinit([Polrev([-1,-1,1/2,1/2]),Polrev([1,1,-1/2,-1/2]),Polrev([-1,-1,1/2,1/2]),Polrev([22,243,-248,-153]),Polrev([-2955,-2876,10239/2,2540])], K);
magma: E := EllipticCurve([K![-1,-1,1/2,1/2],K![1,1,-1/2,-1/2],K![-1,-1,1/2,1/2],K![22,243,-248,-153],K![-2955,-2876,10239/2,2540]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/2a^3-1/2a^2-3a)\) | = | \((1/2a^3-1/2a^2-3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1/2a^3-1/2a^2+a+15)\) | = | \((1/2a^3-1/2a^2-3a)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 29791 \) | = | \(31^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{248777767711978335}{59582} a^{3} + \frac{569082138084219041}{59582} a^{2} + \frac{95151399194580063}{29791} a - \frac{217251974013061967}{29791} \) | ||
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: | $\left(\frac{9}{4} a^{3} + \frac{13}{8} a^{2} - 5 a + \frac{11}{2} : -\frac{17}{4} a^{3} - \frac{19}{2} a^{2} + \frac{37}{8} a + \frac{71}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(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: | \( 170.63134034140379105605341726925923117 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.06644587713377 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^3-1/2a^2-3a)\) | \(31\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
31.4-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.