Base field \(\Q(\zeta_{20})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([5, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,1,1,0]),K([4,-4,-1,1]),K([-3,1,1,0]),K([856,562,-286,-133]),K([-7760,-7692,1792,2304])])
gp: E = ellinit([Polrev([-3,1,1,0]),Polrev([4,-4,-1,1]),Polrev([-3,1,1,0]),Polrev([856,562,-286,-133]),Polrev([-7760,-7692,1792,2304])], K);
magma: E := EllipticCurve([K![-3,1,1,0],K![4,-4,-1,1],K![-3,1,1,0],K![856,562,-286,-133],K![-7760,-7692,1792,2304]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^2-2a+10)\) | = | \((a^3-a^2-3a+2)^{3}\cdot(a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 320 \) | = | \(4^{3}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((160)\) | = | \((a^3-a^2-3a+2)^{10}\cdot(a)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 655360000 \) | = | \(4^{10}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{9285883494578}{5} a^{2} - \frac{12832775369604}{5} \) | ||
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 | $\left(-12 a^{3} - 2 a^{2} + 34 a + 22 : -42 a^{3} - 97 a^{2} + 187 a + 287 : 1\right)$ |
Height | \(0.76325225226125977908508157769865810372\) |
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: | $\left(-\frac{3}{2} a^{3} - 5 a^{2} + \frac{15}{2} a + 11 : \frac{1}{4} a^{3} - a^{2} + \frac{3}{2} a + \frac{7}{4} : 1\right)$ |
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: | \( 0.76325225226125977908508157769865810372 \) | ||
Period: | \( 5.1247543767085294550161754821412523171 \) | ||
Tamagawa product: | \( 8 \) = \(2\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.79882748443601 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-3a+2)\) | \(4\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(3\) | \(10\) | \(0\) |
\((a)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
320.1-c
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q(\sqrt{5}) \) | 2.2.5.1-320.1-b6 |
\(\Q(\sqrt{5}) \) | a curve with conductor norm 6400 (not in the database) |