Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-17424.5-e2
Conductor \((132)\)
Conductor norm \( 17424 \)
CM no
base-change yes: 2112.i4,528.d4
Q-curve yes
Torsion order \( 4 \)
Rank not available

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Show commands for: Magma / SageMath / Pari/GP

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: K = nfinit(a^2 + 2);
 

Weierstrass equation

\( y^2 + a x y = x^{3} + x^{2} + 18 x + 162 \)
magma: E := ChangeRing(EllipticCurve([a, 1, 0, 18, 162]),K);
 
sage: E = EllipticCurve(K, [a, 1, 0, 18, 162])
 
gp: E = ellinit([a, 1, 0, 18, 162],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((132)\) = \( \left(a\right)^{4} \cdot \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(a + 3\right) \cdot \left(a - 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 17424 \) = \( 2^{4} \cdot 3^{2} \cdot 11^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((11290752)\) = \( \left(a\right)^{14} \cdot \left(-a - 1\right)^{6} \cdot \left(a - 1\right)^{6} \cdot \left(a + 3\right)^{2} \cdot \left(a - 3\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 127481080725504 \) = \( 2^{14} \cdot 3^{12} \cdot 11^{4} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{9938375}{176418} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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 not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generators: $\left(-\frac{9}{2} : \frac{9}{4} a : 1\right)$,$\left(-4 a + 2 : -a - 4 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(4\) \(I_{6}^*\) Additive \(1\) \(4\) \(14\) \(2\)
\( \left(-a - 1\right) \) \(3\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(a - 1\right) \) \(3\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(a + 3\right) \) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(a - 3\right) \) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 17424.5-e consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 2112.i4, 528.d4, defined over \(\Q\), so it is also a \(\Q\)-curve.