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([0,1]),K([-9267,-1435]),K([358153,73328])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-9267,-1435]),Polrev([358153,73328])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,1],K![-9267,-1435],K![358153,73328]]);
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: | \((89893440a-270371808)\) | = | \((2)^{5}\cdot(-a-2)^{10}\cdot(-a+3)^{4}\cdot(3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 56254534554387456 \) | = | \(4^{5}\cdot7^{10}\cdot7^{4}\cdot9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{4469851863496388334143}{81352871712} a + \frac{18505449179239697900773}{81352871712} \) | ||
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 | $\left(-5 a + 49 : 91 a + 62 : 1\right)$ |
Height | \(0.98543501754720689620619719235303375877\) |
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: | $\left(\frac{17}{4} a + \frac{107}{2} : -\frac{63}{2} a - \frac{435}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.98543501754720689620619719235303375877 \) | ||
Period: | \( 2.0460859047417760031970980173667998824 \) | ||
Tamagawa product: | \( 40 \) = \(5\cdot2\cdot2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.5391600679301585562821198812679665559 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(4\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
\((-a-2)\) | \(7\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
\((-a+3)\) | \(7\) | \(2\) | \(I_{4}\) | Non-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-h
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.