Properties

Label 2.2.24.1-150.1-e11
Base field \(\Q(\sqrt{6}) \)
Conductor \((-5 a)\)
Conductor norm \( 150 \)
CM no
Base change yes: 30.a1,2880.q1
Q-curve yes
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: x = polygen(QQ); K.<a> = NumberField(x^2 - 6)
 
gp: K = nfinit(a^2 - 6);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
 

Weierstrass equation

\(y^2+xy+y=x^{3}-5334x-150368\)
sage: E = EllipticCurve(K, [1, 0, 1, -5334, -150368])
 
gp: E = ellinit([1, 0, 1, -5334, -150368],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -5334, -150368]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-5 a)\) = \( \left(-a + 2\right) \cdot \left(a + 3\right) \cdot \left(-a - 1\right) \cdot \left(-a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 150 \) = \( 2 \cdot 3 \cdot 5^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((81000)\) = \( \left(-a + 2\right)^{6} \cdot \left(a + 3\right)^{8} \cdot \left(-a - 1\right)^{3} \cdot \left(-a + 1\right)^{3} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6561000000 \) = \( 2^{6} \cdot 3^{8} \cdot 5^{6} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{16778985534208729}{81000} \)
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 $\left(-\frac{1311}{50} a + \frac{1099}{50} : \frac{24909}{500} a - \frac{25353}{250} : 1\right)$
Height \(3.76255927290324\)
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: $\left(-\frac{169}{4} : \frac{165}{8} : 1\right)$
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: \( 3.76255927290324 \)
Period: \( 0.312098809239780 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2^{3}\cdot1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \(1.91760797893023\)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(a + 3\right) \) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\( \left(-a - 1\right) \) \(5\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(-a + 1\right) \) \(5\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 150.1-e consists of curves linked by isogenies of degrees dividing 24.

Base change

This curve is the base change of elliptic curves 30.a1, 2880.q1, defined over \(\Q\), so it is also a \(\Q\)-curve.