Base field 4.4.4752.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -3, -2, 1]))
gp: K = nfinit(Polrev([1, 4, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1,1,0]),K([3,1,-1,0]),K([1,0,0,0]),K([-124,77,-77,0]),K([358,-719,719,0])])
gp: E = ellinit([Polrev([-1,-1,1,0]),Polrev([3,1,-1,0]),Polrev([1,0,0,0]),Polrev([-124,77,-77,0]),Polrev([358,-719,719,0])], K);
magma: E := EllipticCurve([K![-1,-1,1,0],K![3,1,-1,0],K![1,0,0,0],K![-124,77,-77,0],K![358,-719,719,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a+1)\) | = | \((a^3-a^2-4a+1)\cdot(a^2-2a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 33 \) | = | \(3\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((170299a^2-170299a-436478)\) | = | \((a^3-a^2-4a+1)^{2}\cdot(a^2-2a-2)^{20}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 6054749954393040082809 \) | = | \(3^{2}\cdot11^{20}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1081911102879025664}{77812273803} a^{2} - \frac{1081911102879025664}{77812273803} a - \frac{3980255358140970688}{77812273803} \) | ||
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: | \(0\) |
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: | $\left(\frac{15}{2} a^{2} - \frac{15}{2} a - 17 : -\frac{11}{4} a^{2} + \frac{11}{4} a - \frac{21}{4} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 77.218052178530693302460715647477762643 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.12016146005862 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-4a+1)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((a^2-2a-2)\) | \(11\) | \(2\) | \(I_{20}\) | Non-split multiplicative | \(1\) | \(1\) | \(20\) | \(20\) |
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.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
33.1-d
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q(\sqrt{3}) \) | 2.2.12.1-33.2-a3 |
\(\Q(\sqrt{3}) \) | 2.2.12.1-1089.2-f3 |