Properties

Label 2.2.61.1-61.1-a1
Base field \(\Q(\sqrt{61}) \)
Conductor norm \( 61 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{61}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 15 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-15, -1, 1]))
 
gp: K = nfinit(Polrev([-15, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a+1)\) = \((-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 61 \) = \(61\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-61)\) = \((-2a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3721 \) = \(61^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{912673}{61} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(2\)
Generators $\left(1 : 0 : 1\right)$ $\left(-a + 6 : 4 a - 20 : 1\right)$
Heights \(0.079187731362041943636554519799767450798\) \(0.37933208427820208338581717143000910672\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.028470772987258990514108334175982947965 \)
Period: \( 37.616058195513658942657333850408200289 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 1.0969772265936377200602153400409164516 \)
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\((j)_{-}\)
\((-2a+1)\) \(61\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 61.1-a consists of this curve only.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 61.a1
\(\Q\) 3721.a1