Base field 5.5.161121.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 3, -6, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 3, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 3, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([6,7,-11,-2,2]),K([-4,-3,6,1,-1]),K([-1,-4,0,1,0]),K([0,0,0,0,0]),K([0,0,0,0,0])])
gp: E = ellinit([Polrev([6,7,-11,-2,2]),Polrev([-4,-3,6,1,-1]),Polrev([-1,-4,0,1,0]),Polrev([0,0,0,0,0]),Polrev([0,0,0,0,0])], K);
magma: E := EllipticCurve([K![6,7,-11,-2,2],K![-4,-3,6,1,-1],K![-1,-4,0,1,0],K![0,0,0,0,0],K![0,0,0,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a^4-2a^3-11a^2+7a+6)\) | = | \((2a^4-2a^3-11a^2+7a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((3a^4-2a^3-19a^2+6a+10)\) | = | \((2a^4-2a^3-11a^2+7a+6)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 4913 \) | = | \(17^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{10316506}{4913} a^{4} - \frac{15043682}{4913} a^{3} - \frac{53708183}{4913} a^{2} + \frac{56938668}{4913} a + \frac{23313576}{4913} \) | ||
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 | $\left(-a : a^{2} - a - 1 : 1\right)$ |
Height | \(0.017392072162787200884912281727612253694\) |
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: | \( 0.017392072162787200884912281727612253694 \) | ||
Period: | \( 5890.4266976422434122498260696896285539 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.27612147 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2a^4-2a^3-11a^2+7a+6)\) | \(17\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
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
17.1-a
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.