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([-3,-4/3,1/3,1/3]),K([-6,1/3,5/3,-1/3]),K([0,1,0,0]),K([20,-11/3,-49/3,23/3]),K([97,2/3,-275/3,97/3])])
gp: E = ellinit([Polrev([-3,-4/3,1/3,1/3]),Polrev([-6,1/3,5/3,-1/3]),Polrev([0,1,0,0]),Polrev([20,-11/3,-49/3,23/3]),Polrev([97,2/3,-275/3,97/3])], K);
magma: E := EllipticCurve([K![-3,-4/3,1/3,1/3],K![-6,1/3,5/3,-1/3],K![0,1,0,0],K![20,-11/3,-49/3,23/3],K![97,2/3,-275/3,97/3]]);
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: | \( \frac{547727}{2058} a^{3} - \frac{1237028}{1029} a^{2} + \frac{297209}{1029} a + \frac{251089}{98} \) | ||
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 | $\left(-\frac{7}{3} a^{3} + \frac{26}{3} a^{2} - \frac{5}{3} a - 9 : -17 a^{3} + 69 a^{2} - 14 a - 80 : 1\right)$ |
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-d
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.