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([1,-2,0,1]),K([-1,0,0,0]),K([-2,-3,1,1]),K([232,118,-179,-93]),K([-2468,-1281,1810,947])])
gp: E = ellinit([Polrev([1,-2,0,1]),Polrev([-1,0,0,0]),Polrev([-2,-3,1,1]),Polrev([232,118,-179,-93]),Polrev([-2468,-1281,1810,947])], K);
magma: E := EllipticCurve([K![1,-2,0,1],K![-1,0,0,0],K![-2,-3,1,1],K![232,118,-179,-93],K![-2468,-1281,1810,947]]);
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: | \((32a^2-80)\) | = | \((a^3-a^2-3a+2)^{8}\cdot(a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1638400 \) | = | \(4^{8}\cdot5^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1444495316}{5} a^{2} + 1045299956 \) | ||
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 | $\left(6 a^{3} + 12 a^{2} - 6 a - 13 : 38 a^{3} + 72 a^{2} - 56 a - 105 : 1\right)$ | |
Height | \(0.38162612613062988954254078884932905186\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
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 generators: | $\left(3 a^{3} + 4 a^{2} - 7 a - 8 : -4 a^{3} - 7 a^{2} + 7 a + 10 : 1\right)$ | $\left(a + 2 : 7 a^{3} + 14 a^{2} - 9 a - 20 : 1\right)$ |
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: | \( 0.38162612613062988954254078884932905186 \) | ||
Period: | \( 1311.9371204373835404841409234281605932 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 2.79882748443601 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-3a+2)\) | \(4\) | \(2\) | \(I_{1}^{*}\) | Additive | \(1\) | \(3\) | \(8\) | \(0\) |
\((a)\) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
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-b4 |
\(\Q(\sqrt{5}) \) | a curve with conductor norm 6400 (not in the database) |