Base field 4.4.13888.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 6 x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 6, -7, -2, 1]))
gp: K = nfinit(Polrev([9, 6, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 6, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-1/3,-2/3,1/3]),K([0,1/3,2/3,-1/3]),K([-2,-4/3,1/3,1/3]),K([42,-61/3,-32/3,10/3]),K([1802,-2383/3,-1574/3,541/3])])
gp: E = ellinit([Polrev([1,-1/3,-2/3,1/3]),Polrev([0,1/3,2/3,-1/3]),Polrev([-2,-4/3,1/3,1/3]),Polrev([42,-61/3,-32/3,10/3]),Polrev([1802,-2383/3,-1574/3,541/3])], K);
magma: E := EllipticCurve([K![1,-1/3,-2/3,1/3],K![0,1/3,2/3,-1/3],K![-2,-4/3,1/3,1/3],K![42,-61/3,-32/3,10/3],K![1802,-2383/3,-1574/3,541/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/3a^3+4/3a^2-4/3a-3)\) | = | \((1/3a^3-2/3a^2-4/3a+1)\cdot(1/3a^3+1/3a^2-1/3a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 28 \) | = | \(4\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-19/3a^3+38/3a^2+76/3a-25)\) | = | \((1/3a^3-2/3a^2-4/3a+1)\cdot(1/3a^3+1/3a^2-1/3a-1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 470596 \) | = | \(4\cdot7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{557101}{2058} a^{3} + \frac{132200}{1029} a^{2} - \frac{2506865}{1029} a - \frac{1440367}{686} \) | ||
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 | $\left(\frac{2}{3} a^{3} - \frac{10}{3} a^{2} - \frac{2}{3} a + 14 : \frac{13}{3} a^{3} - \frac{50}{3} a^{2} - \frac{40}{3} a + 66 : 1\right)$ |
Height | \(0.078317009194621476981102396331316826989\) |
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: | \( 0.078317009194621476981102396331316826989 \) | ||
Period: | \( 193.37748917792592212587646228190050130 \) | ||
Tamagawa product: | \( 6 \) = \(1\cdot( 2 \cdot 3 )\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.08427716785833 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/3a^3-2/3a^2-4/3a+1)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((1/3a^3+1/3a^2-1/3a-1)\) | \(7\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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
28.2-b
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.