Base field 4.4.13025.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 12 x^{2} + 3 x + 29 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 3, -12, -1, 1]))
gp: K = nfinit(Polrev([29, 3, -12, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 3, -12, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-25/4,1/2,3/2,-1/4]),K([-29/4,1/2,3/2,-1/4]),K([-5,-1,1,0]),K([-42055/4,8953/2,3951/2,-2735/4]),K([-1368261/4,291655/2,128577/2,-89057/4])])
gp: E = ellinit([Polrev([-25/4,1/2,3/2,-1/4]),Polrev([-29/4,1/2,3/2,-1/4]),Polrev([-5,-1,1,0]),Polrev([-42055/4,8953/2,3951/2,-2735/4]),Polrev([-1368261/4,291655/2,128577/2,-89057/4])], K);
magma: E := EllipticCurve([K![-25/4,1/2,3/2,-1/4],K![-29/4,1/2,3/2,-1/4],K![-5,-1,1,0],K![-42055/4,8953/2,3951/2,-2735/4],K![-1368261/4,291655/2,128577/2,-89057/4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/4a^3+1/2a^2-1/2a-7/4)\) | = | \((1/4a^3+1/2a^2-1/2a-7/4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 5 \) | = | \(5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-7/4a^3+13/2a^2+15/2a-103/4)\) | = | \((1/4a^3+1/2a^2-1/2a-7/4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -625 \) | = | \(-5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{16082288916904829}{100} a^{3} + \frac{20149313773541919}{50} a^{2} - \frac{25854498830773729}{50} a - \frac{133033069069667459}{100} \) | ||
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 | $\left(\frac{679}{80} a^{3} - \frac{199}{8} a^{2} - \frac{2311}{40} a + \frac{11203}{80} : \frac{2771}{100} a^{3} - \frac{15773}{200} a^{2} - \frac{7481}{40} a + \frac{86621}{200} : 1\right)$ |
Height | \(2.2320090398831412773471917187207216937\) |
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: | $\left(-\frac{77}{16} a^{3} + \frac{111}{8} a^{2} + \frac{245}{8} a - \frac{1169}{16} : \frac{1}{4} a^{3} - a^{2} + \frac{1}{2} a + \frac{15}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(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: | \( 2.2320090398831412773471917187207216937 \) | ||
Period: | \( 5.7978573157011720964596704616043352896 \) | ||
Tamagawa product: | \( 4 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.81423937825073 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/4a^3+1/2a^2-1/2a-7/4)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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, 4 and 8.
Its isogeny class
5.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.