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([1,0,0,0]),K([25/4,-3/2,-3/2,1/4]),K([5/4,-1/2,-1/2,1/4]),K([721/2,-113,-103,61/2]),K([-12089/4,1871/2,1731/2,-1013/4])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([25/4,-3/2,-3/2,1/4]),Polrev([5/4,-1/2,-1/2,1/4]),Polrev([721/2,-113,-103,61/2]),Polrev([-12089/4,1871/2,1731/2,-1013/4])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![25/4,-3/2,-3/2,1/4],K![5/4,-1/2,-1/2,1/4],K![721/2,-113,-103,61/2],K![-12089/4,1871/2,1731/2,-1013/4]]);
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: | \((3/4a^3-3/2a^2-11/2a+19/4)\) | = | \((1/4a^3+1/2a^2-1/2a-7/4)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{5130041}{20} a^{3} - \frac{7395123}{10} a^{2} - \frac{16865103}{10} a + \frac{78990821}{20} \) | ||
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 | $\left(-\frac{103}{16} a^{3} + \frac{179}{8} a^{2} + \frac{191}{8} a - \frac{1251}{16} : \frac{981}{16} a^{3} - \frac{841}{4} a^{2} - \frac{907}{4} a + \frac{11751}{16} : 1\right)$ |
Height | \(1.1160045199415706386735958593603608468\) |
Torsion structure: | \(\Z/8\Z\) |
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: | $\left(a^{3} - 3 a^{2} - 4 a + 11 : -\frac{27}{4} a^{3} + \frac{45}{2} a^{2} + \frac{51}{2} a - \frac{319}{4} : 1\right)$ |
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: | \( 1.1160045199415706386735958593603608468 \) | ||
Period: | \( 1484.2514728195000566936756381707098341 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 1.81423937825073 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/4a^3+1/2a^2-1/2a-7/4)\) | \(5\) | \(2\) | \(I_{2}\) | 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, 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.