Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-392.1-a4
Conductor \((14 a)\)
Conductor norm \( 392 \)
CM no
base-change yes: 448.e1,56.a1
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

\( y^2 + a x y + a y = x^{3} - x^{2} - 73 x - 211 \)
magma: E := ChangeRing(EllipticCurve([a, -1, a, -73, -211]),K);
sage: E = EllipticCurve(K, [a, -1, a, -73, -211])
gp (2.8): E = ellinit([a, -1, a, -73, -211],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((14 a)\) = \( \left(a\right)^{3} \cdot \left(7\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 392 \) = \( 2^{3} \cdot 49 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((224)\) = \( \left(a\right)^{10} \cdot \left(7\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 50176 \) = \( 2^{10} \cdot 49 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{1443468546}{7} \)
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: $\left(-\frac{2}{9} a - \frac{37}{9} : \frac{70}{27} a + \frac{26}{27} : 1\right)$

Height: 1.0691085767972257

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

Regulator: 1.0691085768

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: $\left(-\frac{9}{2} : \frac{7}{4} a : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(2\) \(III^*\) Additive \(-1\) \(3\) \(10\) \(0\)
\( \left(7\right) \) \(49\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 and 4.
Its isogeny class 392.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 448.e1, 56.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.