Properties

Label 2.2.57.1-228.1-t1
Base field \(\Q(\sqrt{57}) \)
Conductor norm \( 228 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{57}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -1, 1]))
 
gp: K = nfinit(Polrev([-14, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7a-22\right){x}+189a+619\)
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([-22,-7]),K([619,189])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([-22,-7]),Polrev([619,189])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![-22,-7],K![619,189]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-4a+2)\) = \((a-4)\cdot(a+3)\cdot(4a+13)\cdot(10a-43)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 228 \) = \(2\cdot2\cdot3\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((154368a-668160)\) = \((a-4)^{13}\cdot(a+3)^{8}\cdot(4a+13)^{5}\cdot(10a-43)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 9682550784 \) = \(2^{13}\cdot2^{8}\cdot3^{5}\cdot19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1627749557}{4202496} a - \frac{7090865287}{4202496} \)
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: \(1\)
Generator $\left(-1 : 5 a + 17 : 1\right)$
Height \(0.018035926295596278856133544465660315677\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.018035926295596278856133544465660315677 \)
Period: \( 8.9370651386383096434860835924822538301 \)
Tamagawa product: \( 130 \)  =  \(13\cdot2\cdot5\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 5.5509753062216897651777463181736533318 \)
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\((j)_{-}\)
\((a-4)\) \(2\) \(13\) \(I_{13}\) Split multiplicative \(-1\) \(1\) \(13\) \(13\)
\((a+3)\) \(2\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((4a+13)\) \(3\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((10a-43)\) \(19\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 228.1-t consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.