Properties

Base field \(\Q(\sqrt{61}) \)
Label 2.2.61.1-196.1-a6
Conductor \((14)\)
Conductor norm \( 196 \)
CM no
base-change yes: 14.a1,52094.g1
Q-curve yes
Torsion order \( 2 \)
Rank not available

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Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{61}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 15 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 15)
 
gp: K = nfinit(a^2 - a - 15);
 

Weierstrass equation

\( y^2 + x y + y = x^{3} - 2731 x - 55146 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -2731, -55146]),K);
 
sage: E = EllipticCurve(K, [1, 0, 1, -2731, -55146])
 
gp: E = ellinit([1, 0, 1, -2731, -55146],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((14)\) = \( \left(2\right) \cdot \left(7\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 196 \) = \( 4 \cdot 49 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((25088)\) = \( \left(2\right)^{9} \cdot \left(7\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 629407744 \) = \( 4^{9} \cdot 49^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{2251439055699625}{25088} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(-\frac{121}{4} : \frac{117}{8} : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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\((j)_{-}\)
\( \left(2\right) \) \(4\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\( \left(7\right) \) \(49\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 9 and 18.
Its isogeny class 196.1-a consists of curves linked by isogenies of degrees dividing 18.

Base change

This curve is the base-change of elliptic curves 14.a1, 52094.g1, defined over \(\Q\), so it is also a \(\Q\)-curve.