Base field 4.4.13525.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 12 x^{2} + 8 x + 29 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 8, -12, -1, 1]))
gp: K = nfinit(Polrev([29, 8, -12, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 8, -12, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1/5,-7/5,0,1/5]),K([1,1,0,0]),K([-24/5,-7/5,1,1/5]),K([509/5,-203/5,-12,24/5]),K([29/5,7/5,-1,-1/5])])
gp: E = ellinit([Polrev([1/5,-7/5,0,1/5]),Polrev([1,1,0,0]),Polrev([-24/5,-7/5,1,1/5]),Polrev([509/5,-203/5,-12,24/5]),Polrev([29/5,7/5,-1,-1/5])], K);
magma: E := EllipticCurve([K![1/5,-7/5,0,1/5],K![1,1,0,0],K![-24/5,-7/5,1,1/5],K![509/5,-203/5,-12,24/5],K![29/5,7/5,-1,-1/5]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((3/5a^3+a^2-16/5a-22/5)\) | = | \((3/5a^3+a^2-16/5a-22/5)\) |
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: | \((2/5a^3-a^2-29/5a+47/5)\) | = | \((3/5a^3+a^2-16/5a-22/5)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 3125 \) | = | \(5^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{12961}{125} a^{3} - \frac{49314}{125} a^{2} - \frac{11203}{25} a + \frac{275468}{125} \) | ||
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 | $\left(\frac{6}{5} a^{3} - 3 a^{2} - \frac{52}{5} a + \frac{121}{5} : -8 a^{3} + 19 a^{2} + 68 a - 159 : 1\right)$ |
Height | \(0.034210159957605845963609239569575800858\) |
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: | \( 0.034210159957605845963609239569575800858 \) | ||
Period: | \( 431.59691162376433043101781372574847677 \) | ||
Tamagawa product: | \( 5 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.53918761983709 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3/5a^3+a^2-16/5a-22/5)\) | \(5\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(5\) | 5B.4.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
5.1-a
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.