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([-5,1,1]),K([1,0,0]),K([-5,1,1]),K([40,-5,-6]),K([43,-4,-6])])
gp: E = ellinit([Polrev([-5,1,1]),Polrev([1,0,0]),Polrev([-5,1,1]),Polrev([40,-5,-6]),Polrev([43,-4,-6])], K);
magma: E := EllipticCurve([K![-5,1,1],K![1,0,0],K![-5,1,1],K![40,-5,-6],K![43,-4,-6]]);
This is not a global minimal model: it is minimal at all primes except \((2,a^2+a-5)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a)\) | = | \((2,a)\cdot(5,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((784a^2-48a-5680)\) | = | \((2,a)^{4}\cdot(2,a^2+a-5)^{12}\cdot(5,a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -6710886400 \) | = | \(-2^{4}\cdot4^{12}\cdot5^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((2a^2-a-10)\) | = | \((2,a)^{4}\cdot(5,a)^{2}\) |
Minimal discriminant norm: | \( -400 \) | = | \(-2^{4}\cdot5^{2}\) |
j-invariant: | \( \frac{2217577}{400} a^{2} - \frac{6813147}{400} a + \frac{5546677}{400} \) | ||
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 | $\left(-\frac{14}{25} a^{2} - \frac{21}{25} a + \frac{61}{25} : \frac{79}{125} a^{2} + \frac{6}{125} a - \frac{771}{125} : 1\right)$ |
Height | \(0.68881321497971115067121810644350427960\) |
Torsion structure: | \(\Z/4\Z\) |
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: | $\left(a + 3 : -3 a^{2} + 27 : 1\right)$ |
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: | \( 0.68881321497971115067121810644350427960 \) | ||
Period: | \( 244.46143837926587074132877238250457070 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot1\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 5.7096262381248791008264610586228142042 \) | ||
Analytic order of Ш: | \( 1 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((2,a^2+a-5)\) | \(4\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((5,a)\) | \(5\) | \(2\) | \(I_{2}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
10.1-a
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.