Properties

Base field \(\Q(\sqrt{15}) \)
Label 2.2.60.1-14.1-a2
Conductor \((a + 1)\)
Conductor norm \( 14 \)
CM no
base-change no
Q-curve no
Torsion order \( 3 \)
Rank not available

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Base field \(\Q(\sqrt{15}) \)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 15)
 
gp (2.8): K = nfinit(a^2 - 15);
 

Weierstrass equation

\( y^2 + a x y + y = x^{3} + x^{2} + \left(90 a - 343\right) x + 152 a - 585 \)
magma: E := ChangeRing(EllipticCurve([a, 1, 1, 90*a - 343, 152*a - 585]),K);
 
sage: E = EllipticCurve(K, [a, 1, 1, 90*a - 343, 152*a - 585])
 
gp (2.8): E = ellinit([a, 1, 1, 90*a - 343, 152*a - 585],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((a + 1)\) = \( \left(2, a + 1\right) \cdot \left(7, a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 14 \) = \( 2 \cdot 7 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((144 a + 368)\) = \( \left(2, a + 1\right)^{9} \cdot \left(7, a + 1\right)^{3} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 175616 \) = \( 2^{9} \cdot 7^{3} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{80643645}{10976} a - \frac{293363275}{10976} \)
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: \(\Z/3\Z\)
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]
 
Generator: $\left(-5 a + 18 : -32 a + 126 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(2, a + 1\right) \) \(2\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\( \left(7, a + 1\right) \) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 14.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

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