Base field \(\Q(\sqrt{53}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 13 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-13, -1, 1]))
gp: K = nfinit(Polrev([-13, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([1,-1]),K([1,0]),K([-205,42]),K([1027,-273])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([1,0]),Polrev([-205,42]),Polrev([1027,-273])], K);
magma: E := EllipticCurve([K![1,1],K![1,-1],K![1,0],K![-205,42],K![1027,-273]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((42)\) | = | \((2)\cdot(-a-2)\cdot(-a+3)\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 1764 \) | = | \(4\cdot7\cdot7\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1512a+11592)\) | = | \((2)^{3}\cdot(-a-2)\cdot(-a+3)^{4}\cdot(3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 87127488 \) | = | \(4^{3}\cdot7\cdot7^{4}\cdot9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{19062050559161}{9604} a + \frac{1420525285468939}{172872} \) | ||
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: | \(2\) | |
Generators | $\left(9 : -8 a - 16 : 1\right)$ | $\left(-\frac{94}{49} a + \frac{149}{49} : \frac{45}{343} a + \frac{3428}{343} : 1\right)$ |
Heights | \(0.81547456993931967710174557768997438943\) | \(1.1989551854399631418554029276968568555\) |
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);
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Torsion generator: | $\left(-\frac{7}{4} a + \frac{7}{2} : \frac{73}{8} : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.96279102899506346133802208245673120736 \) | ||
Period: | \( 8.2673900069417899124246940928581421078 \) | ||
Tamagawa product: | \( 8 \) = \(1\cdot1\cdot2^{2}\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 8.7468667950820089810574803331190716068 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(4\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((-a-2)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((-a+3)\) | \(7\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((3)\) | \(9\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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
1764.1-d
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.