Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-1444.2-b2
Conductor \((38)\)
Conductor norm \( 1444 \)
CM no
base-change yes: 1862.f2,38.b2
Q-curve yes
Torsion order \( 5 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{-7}) \)

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

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

Weierstrass equation

\( y^2 + x y + y = x^{3} + x^{2} + 1 \)
magma: E := ChangeRing(EllipticCurve([1, 1, 1, 0, 1]),K);
sage: E = EllipticCurve(K, [1, 1, 1, 0, 1])
gp (2.8): E = ellinit([1, 1, 1, 0, 1],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((38)\) = \( \left(a\right) \cdot \left(-a + 1\right) \cdot \left(19\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 1444 \) = \( 2^{2} \cdot 361 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((608)\) = \( \left(a\right)^{5} \cdot \left(-a + 1\right)^{5} \cdot \left(19\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 369664 \) = \( 2^{10} \cdot 361 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{1}{608} \)
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: \( 0 \)
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 1

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

Torsion subgroup

Structure: \(\Z/5\Z\)
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: $\left(-1 : -1 : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\( \left(-a + 1\right) \) \(2\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\( \left(19\right) \) \(361\) \(1\) \(I_{1}\) 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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 1444.2-b consists of curves linked by isogenies of degree5.

Base change

This curve is the base-change of elliptic curves 1862.f2, 38.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.