Base field 3.3.1957.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 9 x + 10 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([10, -9, -1, 1]))
gp: K = nfinit(Polrev([10, -9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0]),K([-6,0,1]),K([0,1,0]),K([59066077,-32872327,-17197107]),K([276022222850,-153615979044,-80363961659])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([-6,0,1]),Polrev([0,1,0]),Polrev([59066077,-32872327,-17197107]),Polrev([276022222850,-153615979044,-80363961659])], K);
magma: E := EllipticCurve([K![1,1,0],K![-6,0,1],K![0,1,0],K![59066077,-32872327,-17197107],K![276022222850,-153615979044,-80363961659]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((11,a-4)\) | = | \((11,a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-16a^2+289a-9)\) | = | \((11,a-4)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 214358881 \) | = | \(11^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{5510373784198200}{214358881} a^{2} - \frac{22266438999717392}{214358881} a + \frac{18120112986554473}{214358881} \) | ||
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 | $\left(-751 a^{2} - 1435 a + 2580 : 30617 a^{2} + 58524 a - 105160 : 1\right)$ | |
Height | \(1.5419092243197487345004516942948840686\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(1453 a^{2} + 2778 a - 4990 : -2842 a^{2} - 5433 a + 9760 : 1\right)$ | $\left(-535 a^{2} - 1022 a + 1838 : 1046 a^{2} + 1999 a - 3594 : 1\right)$ |
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: | \( 1.5419092243197487345004516942948840686 \) | ||
Period: | \( 22.122774189315788911492857280200936877 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.1566294317998090015866604447952603614 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((11,a-4)\) | \(11\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
11.1-b
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.