Base field \(\Q(\sqrt{301}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 75 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-75, -1, 1]))
gp: K = nfinit(Polrev([-75, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-75, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([-474,76]),K([2477,-200])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-474,76]),Polrev([2477,-200])], K);
magma: E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![-474,76],K![2477,-200]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a-8)\) | = | \((-a-8)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 3 \) | = | \(3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-7a+64)\) | = | \((-a-8)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -27 \) | = | \(-3^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{5404}{27} a + \frac{9443}{9} \) | ||
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 | $\left(-3 a + 19 : 8 a - 30 : 1\right)$ |
Height | \(0.55461175809995551714340623037574317926\) |
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: | \( 0.55461175809995551714340623037574317926 \) | ||
Period: | \( 20.975784407081197792825559850660809541 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 4.0232339353974142487790327091040881203 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a-8)\) | \(3\) | \(3\) | \(I_{3}\) | 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 |
---|---|
\(3\) | 3Ns |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 3.2-a 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.