Properties

Label 2.2.24.1-150.1-c2
Base field \(\Q(\sqrt{6}) \)
Conductor norm \( 150 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(84a-213\right){x}+710a-1749\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,1]),K([-213,84]),K([-1749,710])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,1]),Polrev([-213,84]),Polrev([-1749,710])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,1],K![-213,84],K![-1749,710]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-5a)\) = \((-a+2)\cdot(a+3)\cdot(-a-1)\cdot(-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 150 \) = \(2\cdot3\cdot5\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-9600a+78600)\) = \((-a+2)^{6}\cdot(a+3)^{2}\cdot(-a-1)^{2}\cdot(-a+1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5625000000 \) = \(2^{6}\cdot3^{2}\cdot5^{2}\cdot5^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2583922302027}{1562500} a + \frac{38016780418783}{9375000} \)
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: \(0\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(2 a - \frac{29}{4} : -\frac{3}{2} a + \frac{25}{8} : 1\right)$ $\left(-5 a + 11 : 2 a - 6 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 3.2645143971183417723444266960760034915 \)
Tamagawa product: \( 48 \)  =  \(( 2 \cdot 3 )\cdot2\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.9990986327273471771201421938218330606 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((a+3)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-a-1)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-a+1)\) \(5\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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.
Its isogeny class 150.1-c consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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