Properties

Base field \(\Q(\sqrt{29}) \)
Label 2.2.29.1-28.1-b1
Conductor \((-2 a)\)
Conductor norm \( 28 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field \(\Q(\sqrt{29}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 7 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 7)
gp (2.8): K = nfinit(a^2 - a - 7);

Weierstrass equation

\( y^2 + \left(a + 1\right) x y + a y = x^{3} + \left(a + 1\right) x^{2} + \left(3 a + 5\right) x + 22 a + 47 \)
magma: E := ChangeRing(EllipticCurve([a + 1, a + 1, a, 3*a + 5, 22*a + 47]),K);
sage: E = EllipticCurve(K, [a + 1, a + 1, a, 3*a + 5, 22*a + 47])
gp (2.8): E = ellinit([a + 1, a + 1, a, 3*a + 5, 22*a + 47],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-2 a)\) = \( \left(2\right) \cdot \left(-a\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 28 \) = \( 4 \cdot 7 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((960 a + 3584)\) = \( \left(2\right)^{6} \cdot \left(-a\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 9834496 \) = \( 4^{6} \cdot 7^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{198387025}{153664} a - \frac{79186783}{19208} \)
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 not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

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\((j)_{-}\)
\( \left(-a\right) \) \(7\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\( \left(2\right) \) \(4\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

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 28.1-b 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.