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([0,1,0]),K([1,0,0]),K([-6,1,1]),K([-6677,10355,-3932]),K([-1166954,1354261,-284267])])
gp: E = ellinit([Polrev([0,1,0]),Polrev([1,0,0]),Polrev([-6,1,1]),Polrev([-6677,10355,-3932]),Polrev([-1166954,1354261,-284267])], K);
magma: E := EllipticCurve([K![0,1,0],K![1,0,0],K![-6,1,1],K![-6677,10355,-3932],K![-1166954,1354261,-284267]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2,a)\) | = | \((2,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 2 \) | = | \(2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-145a^2+671a-1582)\) | = | \((2,a)^{32}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 4294967296 \) | = | \(2^{32}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((7a^2+24a-172)\) | = | \((2,a)^{20}\) |
Minimal discriminant norm: | \( 1048576 \) | = | \(2^{20}\) |
j-invariant: | \( \frac{1208201331713038647}{1048576} a^{2} - \frac{4882016462354803125}{1048576} a + \frac{3972767044836889003}{1048576} \) | ||
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 | $\left(-\frac{68}{9} a^{2} + \frac{212}{9} a - \frac{155}{9} : -\frac{2011}{27} a^{2} - 84 a + \frac{1727}{9} : 1\right)$ | |
Height | \(2.5568170363389331088681900905562104088\) | |
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: | $\left(18 a^{2} - 41 a + 23 : 11 a^{2} - 93 a + 93 : 1\right)$ | $\left(-a^{2} + 36 a - 40 : -18 a^{2} + 24 a - 2 : 1\right)$ |
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: | \( 2.5568170363389331088681900905562104088 \) | ||
Period: | \( 19.962223131496859480918550133573749325 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.7306306993003446798793419761692342849 \) | ||
Analytic order of Ш: | \( 4 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(2\) | \(I_{20}\) | Non-split multiplicative | \(1\) | \(1\) | \(20\) | \(20\) |
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 |
\(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
2.1-b
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.