Base field 4.4.16317.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([1, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([1,-1,0,0]),K([-1,-5,0,1]),K([-18,-113,-42,30]),K([-1227,-7166,-1117,1419])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([1,-1,0,0]),Polrev([-1,-5,0,1]),Polrev([-18,-113,-42,30]),Polrev([-1227,-7166,-1117,1419])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![1,-1,0,0],K![-1,-5,0,1],K![-18,-113,-42,30],K![-1227,-7166,-1117,1419]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3+2a^2+3a-3)\) | = | \((-a^3+2a^2+3a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 7 \) | = | \(7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^2-2a-1)\) | = | \((-a^3+2a^2+3a-3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 7 \) | = | \(7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{115785963708214013}{7} a^{3} + \frac{433882723088110564}{7} a^{2} - \frac{293930821662511777}{7} a - \frac{66390659360476439}{7} \) | ||
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 | $\left(-\frac{1353}{25} a^{3} + \frac{901}{20} a^{2} + \frac{6699}{25} a + \frac{4779}{100} : -\frac{194087}{200} a^{3} + \frac{796871}{1000} a^{2} + \frac{4823009}{1000} a + \frac{827533}{1000} : 1\right)$ |
Height | \(2.2891192591058210554331616697847655162\) |
Torsion structure: | \(\Z/3\Z\) |
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: | $\left(-6 a^{3} + 6 a^{2} + \frac{85}{3} a + \frac{13}{3} : \frac{286}{9} a^{3} - \frac{245}{9} a^{2} - 157 a - \frac{233}{9} : 1\right)$ |
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: | \( 2.2891192591058210554331616697847655162 \) | ||
Period: | \( 285.76834250970319386411585129547527646 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 2.27603984269792 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^3+2a^2+3a-3)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
7.1-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.