Base field \(\Q(\sqrt{69}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 17 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-17, -1, 1]))
gp: K = nfinit(Polrev([-17, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([1,1]),K([-191,38]),K([-1585,339])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,-1]),Polrev([1,1]),Polrev([-191,38]),Polrev([-1585,339])], K);
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![1,1],K![-191,38],K![-1585,339]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-6a+26)\) | = | \((2)\cdot(-3a+13)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 92 \) | = | \(4\cdot23\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-24a+104)\) | = | \((2)^{3}\cdot(-3a+13)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -1472 \) | = | \(-4^{3}\cdot23\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{30308013}{184} a + \frac{70457445}{92} \) | ||
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 | $\left(\frac{4}{3} a - 7 : \frac{16}{9} a - \frac{80}{9} : 1\right)$ |
Height | \(0.41281776061617713205124585341151509404\) |
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: | \( 0.41281776061617713205124585341151509404 \) | ||
Period: | \( 16.614122493233420745221530883237581372 \) | ||
Tamagawa product: | \( 3 \) = \(3\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 4.9540739694815687589668731301063424030 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(4\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
\((-3a+13)\) | \(23\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 92.1-e 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.