Properties

Label 2.2.24.1-870.3-g2
Base field \(\Q(\sqrt{6}) \)
Conductor norm \( 870 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{6}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 6 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
 
gp: K = nfinit(Polrev([-6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2283a-5588\right){x}+92480a+226530\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([-5588,-2283]),K([226530,92480])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([-5588,-2283]),Polrev([226530,92480])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![-5588,-2283],K![226530,92480]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-19a+36)\) = \((-a+2)\cdot(a+3)\cdot(-a-1)\cdot(3a+5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 870 \) = \(2\cdot3\cdot5\cdot29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((13068a-23112)\) = \((-a+2)^{5}\cdot(a+3)^{6}\cdot(-a-1)^{2}\cdot(3a+5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -490471200 \) = \(-2^{5}\cdot3^{6}\cdot5^{2}\cdot29^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{164321962690019249}{4541400} a + \frac{201252481903271483}{2270700} \)
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 $\left(14 a + 34 : -45 a - 110 : 1\right)$
Height \(0.10024983669587499909455784545875207517\)
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);
 
Torsion generator: $\left(\frac{25}{2} a + \frac{121}{4} : -\frac{171}{8} a - \frac{421}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.10024983669587499909455784545875207517 \)
Period: \( 5.1016728375217879772841292022505395234 \)
Tamagawa product: \( 120 \)  =  \(5\cdot( 2 \cdot 3 )\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 6.2638580587346251186724879116193060936 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((a+3)\) \(3\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((-a-1)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((3a+5)\) \(29\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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.
Its isogeny class 870.3-g consists of curves linked by isogenies of degree 2.

Base change

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

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