Base field 4.4.17069.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} - 4 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -4, -8, -1, 1]))
gp: K = nfinit(Polrev([3, -4, -8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -4, -8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-7,-1,1]),K([5,7,1,-1]),K([-3,-6,-1,1]),K([55,58,-1,-6]),K([105,154,24,-22])])
gp: E = ellinit([Polrev([-3,-7,-1,1]),Polrev([5,7,1,-1]),Polrev([-3,-6,-1,1]),Polrev([55,58,-1,-6]),Polrev([105,154,24,-22])], K);
magma: E := EllipticCurve([K![-3,-7,-1,1],K![5,7,1,-1],K![-3,-6,-1,1],K![55,58,-1,-6],K![105,154,24,-22]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3+2a^2+5a)\) | = | \((a)\cdot(-a^2+2a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 9 \) | = | \(3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((2a^3-11a-9)\) | = | \((a)\cdot(-a^2+2a+5)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 81 \) | = | \(3\cdot3^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{16679643524}{27} a^{3} - \frac{5013645460}{3} a^{2} - \frac{56448461584}{27} a + \frac{9792839605}{9} \) | ||
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 | $\left(a^{3} - a^{2} - 7 a - 2 : a^{3} - 2 a^{2} - 10 a - 7 : 1\right)$ |
Height | \(0.43407740520619932413661024539977379915\) |
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: | $\left(a^{3} - 2 a^{2} - 5 a - 3 : a^{3} - 10 a - 9 : 1\right)$ |
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: | \( 0.43407740520619932413661024539977379915 \) | ||
Period: | \( 328.67574408694071929868731601168027436 \) | ||
Tamagawa product: | \( 3 \) = \(1\cdot3\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.27606153532055 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((-a^2+2a+5)\) | \(3\) | \(3\) | \(I_{3}\) | 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 |
---|---|
\(2\) | 2B |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
9.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.