Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 15 x^{2} + 16 x + 29 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 16, -15, -2, 1]))
gp: K = nfinit(Polrev([29, 16, -15, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 16, -15, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-21/23,63/23,6/23,-4/23]),K([104/23,33/23,-10/23,-1/23]),K([1/23,43/23,3/23,-2/23]),K([1153/23,-147/23,-106/23,17/23]),K([-3577/23,1922/23,424/23,-160/23])])
gp: E = ellinit([Polrev([-21/23,63/23,6/23,-4/23]),Polrev([104/23,33/23,-10/23,-1/23]),Polrev([1/23,43/23,3/23,-2/23]),Polrev([1153/23,-147/23,-106/23,17/23]),Polrev([-3577/23,1922/23,424/23,-160/23])], K);
magma: E := EllipticCurve([K![-21/23,63/23,6/23,-4/23],K![104/23,33/23,-10/23,-1/23],K![1/23,43/23,3/23,-2/23],K![1153/23,-147/23,-106/23,17/23],K![-3577/23,1922/23,424/23,-160/23]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-4/23a^3+6/23a^2+63/23a-21/23)\) | = | \((-4/23a^3+6/23a^2+63/23a-21/23)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4/23a^3+6/23a^2+63/23a-21/23)\) | = | \((-4/23a^3+6/23a^2+63/23a-21/23)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 29 \) | = | \(29\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{2260418171}{667} a^{3} - \frac{11897954852}{667} a^{2} + \frac{4926111247}{667} a + \frac{692520183}{23} \) | ||
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 | $\left(\frac{6}{23} a^{3} - \frac{9}{23} a^{2} - \frac{83}{23} a + \frac{89}{23} : -\frac{13}{23} a^{3} + \frac{31}{23} a^{2} + \frac{153}{23} a - \frac{235}{23} : 1\right)$ |
Height | \(0.18325482942207945916954487270676713506\) |
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: | \( 0.18325482942207945916954487270676713506 \) | ||
Period: | \( 501.70341450493858853440672853273254364 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.62684496130221 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-4/23a^3+6/23a^2+63/23a-21/23)\) | \(29\) | \(1\) | \(I_{1}\) | 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
29.2-a
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.