Properties

Base field \(\Q(\sqrt{29}) \)
Label 2.2.29.1-20.2-a1
Conductor \((-2 a + 4)\)
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

\( y^2 + \left(a + 1\right) x y = x^{3} + \left(142 a - 449\right) x + 1521 a - 4857 \)
magma: E := ChangeRing(EllipticCurve([a + 1, 0, 0, 142*a - 449, 1521*a - 4857]),K);
 
sage: E = EllipticCurve(K, [a + 1, 0, 0, 142*a - 449, 1521*a - 4857])
 
gp (2.8): E = ellinit([a + 1, 0, 0, 142*a - 449, 1521*a - 4857],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-2 a + 4)\) = \( \left(2\right) \cdot \left(-a + 2\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 20 \) = \( 4 \cdot 5 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((-512 a + 1024)\) = \( \left(2\right)^{9} \cdot \left(-a + 2\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 1310720 \) = \( 4^{9} \cdot 5 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{19984640951}{640} a - \frac{175841574349}{2560} \)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(2\right) \) \(4\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 20.2-a consists of curves linked by isogenies of degree 3.

Base change

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