Base field 4.4.4913.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, 1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3/2,3,1,-1/2]),K([3/2,3,0,-1/2]),K([-1/2,-2,0,1/2]),K([-46,229,-295,49]),K([2353/2,-245,-5454,4121/2])])
gp: E = ellinit([Polrev([-3/2,3,1,-1/2]),Polrev([3/2,3,0,-1/2]),Polrev([-1/2,-2,0,1/2]),Polrev([-46,229,-295,49]),Polrev([2353/2,-245,-5454,4121/2])], K);
magma: E := EllipticCurve([K![-3/2,3,1,-1/2],K![3/2,3,0,-1/2],K![-1/2,-2,0,1/2],K![-46,229,-295,49],K![2353/2,-245,-5454,4121/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((1/2a^3-a^2-2a+5/2)\cdot(-a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 16 \) | = | \(4\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((4a^3-40a-20)\) | = | \((1/2a^3-a^2-2a+5/2)^{6}\cdot(-a-1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 262144 \) | = | \(4^{6}\cdot4^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{653762688677050897}{64} a^{3} + \frac{653762688677050897}{64} a^{2} + \frac{3268813443385254485}{64} a + \frac{1020884965413408563}{64} \) | ||
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: | $\left(\frac{33}{8} a^{3} - \frac{23}{4} a^{2} - \frac{53}{4} a + \frac{45}{8} : \frac{3}{4} a^{3} - \frac{13}{4} a^{2} - 12 a + \frac{59}{8} : 1\right)$ |
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: | \( 6.5089377254786706738534646532954055054 \) | ||
Tamagawa product: | \( 18 \) = \(( 2 \cdot 3 )\cdot3\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.67151100189927 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^3-a^2-2a+5/2)\) | \(4\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
\((-a-1)\) | \(4\) | \(3\) | \(I_{3}\) | 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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6, 8, 12 and 24.
Its isogeny class
16.1-b
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q(\sqrt{17}) \) | 2.2.17.1-4.1-a6 |
\(\Q(\sqrt{17}) \) | a curve with conductor norm 1156 (not in the database) |