Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-3721.1-b1
Conductor \((61)\)
Conductor norm \( 3721 \)
CM no
base-change yes: 61.a1,1525.b1
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{5}) \)

Generator \(\phi\), with minimal polynomial \( x^{2} - x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<phi> := NumberField(R![-1, -1, 1]);
sage: x = polygen(QQ); K.<phi> = NumberField(x^2 - x - 1)
gp (2.8): K = nfinit(phi^2 - phi - 1);

Weierstrass equation

\( y^2 + x y = x^{3} - 2 x + 1 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -2, 1]),K);
sage: E = EllipticCurve(K, [1, 0, 0, -2, 1])
gp (2.8): E = ellinit([1, 0, 0, -2, 1],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((61)\) = \( \left(7 \phi - 3\right) \cdot \left(7 \phi - 4\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 3721 \) = \( 61^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((61)\) = \( \left(7 \phi - 3\right) \cdot \left(7 \phi - 4\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 3721 \) = \( 61^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{912673}{61} \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 2 \)
magma: Rank(E);
sage: E.rank()

Generators: $\left(-\phi : \phi : 1\right)$, $\left(\phi - 1 : 0 : 1\right)$

Heights: 0.3975049457051695, 0.3975049457051695

magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 0.029909840656

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: Trivial
magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(7 \phi - 3\right) \) \(61\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(7 \phi - 4\right) \) \(61\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 3721.1-b consists of this curve only.

Base change

This curve is the base-change of elliptic curves 61.a1, 1525.b1, defined over \(\Q\), so it is also a \(\Q\)-curve.