Properties

Label 2.2.24.1-150.1-e1
Base field \(\Q(\sqrt{6}) \)
Conductor \((-5 a)\)
Conductor norm \( 150 \)
CM no
Base change no
Q-curve yes
Torsion order \( 6 \)
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

\(y^2+xy+y=x^{3}+\left(885a-2449\right)x-24162a+61154\)
sage: E = EllipticCurve(K, [1, 0, 1, 885*a - 2449, -24162*a + 61154])
 
gp: E = ellinit([1, 0, 1, 885*a - 2449, -24162*a + 61154],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 1, 885*a - 2449, -24162*a + 61154]),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: \((-9825 a - 7200)\) = \( \left(-a + 2\right) \cdot \left(a + 3\right)^{3} \cdot \left(-a - 1\right)^{8} \cdot \left(-a + 1\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 527343750 \) = \( 2 \cdot 3^{3} \cdot 5^{10} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{8889588234944170142939}{7031250} a + \frac{1209719733278025672168}{390625} \)
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 $\left(\frac{9}{2} a + \frac{301}{6} : \frac{3325}{36} a + \frac{1237}{6} : 1\right)$
Height \(2.50837284860216\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-5 a + 28 : 8 a + 11 : 1\right)$
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: \( 2.50837284860216 \)
Period: \( 5.61777856631605 \)
Tamagawa product: \( 12 \)  =  \(1\cdot3\cdot2\cdot2\)
Torsion order: \(6\)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(2\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(a + 3\right) \) \(3\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\( \left(-a - 1\right) \) \(5\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\( \left(-a + 1\right) \) \(5\) \(2\) \(I_{2}\) 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
\(3\) 3B.1.1

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 not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.