Base field \(\Q(\sqrt{19}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, 0, 1]))
gp: K = nfinit(Polrev([-19, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([0,1]),K([-2875005,659571]),K([2652314563,-608482693])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([0,1]),Polrev([-2875005,659571]),Polrev([2652314563,-608482693])], K);
magma: E := EllipticCurve([K![1,0],K![1,0],K![0,1],K![-2875005,659571],K![2652314563,-608482693]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((3a)\) | = | \((-a-4)\cdot(-a+4)\cdot(a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 171 \) | = | \(3\cdot3\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1701a-9234)\) | = | \((-a-4)^{5}\cdot(-a+4)^{8}\cdot(a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 30292137 \) | = | \(3^{5}\cdot3^{8}\cdot19\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{12364905437067631}{124659} a - \frac{2836704000281954}{6561} \) | ||
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 | $\left(-\frac{1496}{9} a + \frac{6518}{9} : -\frac{10793}{27} a + \frac{46991}{27} : 1\right)$ |
Height | \(1.6607444899245993771950212302113705866\) |
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: | \( 1.6607444899245993771950212302113705866 \) | ||
Period: | \( 3.6049981340922071137779633069962206045 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot2\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.7470151820124340897666118171392182666 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a-4)\) | \(3\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
\((-a+4)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((a)\) | \(19\) | \(1\) | \(I_{1}\) | 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 171.1-c 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.