Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-158.4-a1
Conductor \((a - 13)\)
Conductor norm \( 158 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((a - 13)\) = \( \left(-a + 1\right) \cdot \left(6 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 158 \) = \( 2 \cdot 79 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((15 a - 37)\) = \( \left(-a + 1\right)^{4} \cdot \left(6 a + 1\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 1264 \) = \( 2^{4} \cdot 79 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{226873}{1264} a - \frac{264187}{632} \)
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: $\left(1 : -a - 1 : 1\right)$

Height: 0.02018433583243251

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

Regulator: 0.0201843358324

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

Torsion subgroup

Structure: Trivial
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]

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\((j)_{-}\)
\( \left(-a + 1\right) \) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\( \left(6 a + 1\right) \) \(79\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 158.4-a consists of this curve only.

Base change

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