Base field \(\Q(\sqrt{161}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 40 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-40, -1, 1]))
gp: K = nfinit(Polrev([-40, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-40, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([1,0]),K([187,18]),K([1695,267])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([1,0]),Polrev([187,18]),Polrev([1695,267])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![1,0],K![187,18],K![1695,267]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-5a-29)\) | = | \((-a+7)\cdot(-32a+219)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-a-3)\) | = | \((-a+7)^{2}\cdot(-32a+219)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -28 \) | = | \(-2^{2}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{3389}{28} a + \frac{9710}{7} \) | ||
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 | $\left(-\frac{9}{25} a - \frac{181}{25} : \frac{582}{125} a + \frac{1463}{125} : 1\right)$ | $\left(-\frac{1}{4} a - \frac{235}{36} : \frac{1207}{216} a + \frac{3745}{216} : 1\right)$ |
Heights | \(1.2574623635838583998323407480371261257\) | \(2.9551231594850382556995020782782222784\) |
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: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.12505515524826099674856842823582233261 \) | ||
Period: | \( 31.809296674201043778998607047618203376 \) | ||
Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.5080299324327744356800218787162916335 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+7)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((-32a+219)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
14.1-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.