Properties

Base field \(\Q(\sqrt{29}) \)
Label 2.2.29.1-20.1-b1
Conductor \((-2 a - 2)\)
Conductor norm \( 20 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-2 a - 2)\) = \( \left(2\right) \cdot \left(-a - 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 20 \) = \( 4 \cdot 5 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((118 a - 702)\) = \( \left(2\right) \cdot \left(-a - 1\right)^{7} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 312500 \) = \( 4 \cdot 5^{7} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{119629571}{156250} a + \frac{805426}{78125} \)
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\((j)_{-}\)
\( \left(-a - 1\right) \) \(5\) \(1\) \(I_{7}\) Non-split multiplicative \(1\) \(1\) \(7\) \(7\)
\( \left(2\right) \) \(4\) \(1\) \(I_{1}\) 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 20.1-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.