Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-1922.1-c5
Conductor \((31 a)\)
Conductor norm \( 1922 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}) \)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 2)
 
gp (2.8): K = nfinit(a^2 - 2);
 

Weierstrass equation

\( y^2 + x y + y = x^{3} - x^{2} + \left(-165 a - 111\right) x - 818 a - 1207 \)
magma: E := ChangeRing(EllipticCurve([1, -1, 1, -165*a - 111, -818*a - 1207]),K);
 
sage: E = EllipticCurve(K, [1, -1, 1, -165*a - 111, -818*a - 1207])
 
gp (2.8): E = ellinit([1, -1, 1, -165*a - 111, -818*a - 1207],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((31 a)\) = \( \left(a\right) \cdot \left(-4 a - 1\right) \cdot \left(4 a - 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 1922 \) = \( 2 \cdot 31^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((5488271 a - 41224978)\) = \( \left(a\right) \cdot \left(-4 a - 1\right)^{2} \cdot \left(4 a - 1\right)^{8} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 1639256573961602 \) = \( 2 \cdot 31^{10} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{194892158021341473}{1705782074882} a + \frac{139281368709237480}{852891037441} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 1 \)
magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(\frac{41}{2} a + \frac{63}{2} : -\frac{643}{4} a - \frac{491}{2} : 1\right)$

Height: 4.073673831996567

magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 4.073673832

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generator: $\left(-4 a - \frac{21}{4} : 2 a + \frac{17}{8} : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a\right) \) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(-4 a - 1\right) \) \(31\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(4 a - 1\right) \) \(31\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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, 4 and 8.
Its isogeny class 1922.1-c consists of curves linked by isogenies of degrees dividing 8.

Base change

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