Properties

Base field \(\Q(\sqrt{14}) \)
Label 2.2.56.1-112.1-b5
Conductor \((-8 a + 28)\)
Conductor norm \( 112 \)
CM no
base-change yes: 112.b1,3136.q1
Q-curve yes
Torsion order \( 4 \)
Rank not available

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Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{14}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 14)
gp (2.8): K = nfinit(a^2 - 14);

Weierstrass equation

\( y^2 + a x y = x^{3} + x^{2} + \left(8970 a - 33556\right) x + 907960 a - 3397272 \)
magma: E := ChangeRing(EllipticCurve([a, 1, 0, 8970*a - 33556, 907960*a - 3397272]),K);
sage: E = EllipticCurve(K, [a, 1, 0, 8970*a - 33556, 907960*a - 3397272])
gp (2.8): E = ellinit([a, 1, 0, 8970*a - 33556, 907960*a - 3397272],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-8 a + 28)\) = \( \left(-a + 4\right)^{4} \cdot \left(-2 a + 7\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 112 \) = \( 2^{4} \cdot 7 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((224)\) = \( \left(-a + 4\right)^{10} \cdot \left(-2 a + 7\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 50176 \) = \( 2^{10} \cdot 7^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{1443468546}{7} \)
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: \(\Z/2\Z\times\Z/2\Z\)
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]
Generators: $\left(20 a - \frac{153}{2} : \frac{153}{4} a - 140 : 1\right)$,$\left(-40 a + 148 : -74 a + 280 : 1\right)$
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\((j)_{-}\)
\( \left(-a + 4\right) \) \(2\) \(4\) \(I_{2}^*\) Additive \(-1\) \(4\) \(10\) \(0\)
\( \left(-2 a + 7\right) \) \(7\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 112.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 112.b1, 3136.q1, defined over \(\Q\), so it is also a \(\Q\)-curve.