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([-267,320,-72]),K([-4094,5315,-1499])])
gp: E = ellinit([Polrev([0,1,0]),Polrev([1,0,0]),Polrev([-6,1,1]),Polrev([-267,320,-72]),Polrev([-4094,5315,-1499])], K);
magma: E := EllipticCurve([K![0,1,0],K![1,0,0],K![-6,1,1],K![-267,320,-72],K![-4094,5315,-1499]]);
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: | \((9a^2-3a-58)\) | = | \((2,a)^{14}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -16384 \) | = | \(-2^{14}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((a-2)\) | = | \((2,a)^{2}\) |
Minimal discriminant norm: | \( -4 \) | = | \(-2^{2}\) |
j-invariant: | \( -\frac{303439985770557}{4} a^{2} - \frac{39283729367165}{4} a + \frac{2686590421306587}{4} \) | ||
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 | $\left(6 a - 7 : -6 a^{2} - 3 a + 13 : 1\right)$ |
Height | \(1.0227268145355732435472760362224841635\) |
Torsion structure: | \(\Z/2\Z\) |
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 generator: | $\left(-\frac{1}{4} a^{2} + 6 a - 7 : -\frac{27}{8} a^{2} + \frac{33}{8} a + \frac{7}{4} : 1\right)$ |
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: | \( 1.0227268145355732435472760362224841635 \) | ||
Period: | \( 12.476389457185537175574093833483593328 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
\(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 5, 10 and 20.
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.