Properties

Base field 4.4.2624.1
Label 4.4.2624.1-49.4-a3
Conductor \((49,a^{3} - a^{2} - 4 a + 1)\)
Conductor norm \( 49 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field 4.4.2624.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 - 3*x^2 + 2*x + 1)
gp (2.8): K = nfinit(a^4 - 2*a^3 - 3*a^2 + 2*a + 1);

Weierstrass equation

\( y^2 + \left(a^{3} - 2 a^{2} - 2 a + 1\right) x y + \left(a^{2} - 2 a\right) y = x^{3} + \left(-a^{3} + 2 a^{2} + 3 a - 2\right) x^{2} + \left(4 a^{3} - 24 a^{2} - 22 a - 8\right) x + 4 a^{3} + 39 a^{2} + 126 a + 47 \)
magma: E := ChangeRing(EllipticCurve([a^3 - 2*a^2 - 2*a + 1, -a^3 + 2*a^2 + 3*a - 2, a^2 - 2*a, 4*a^3 - 24*a^2 - 22*a - 8, 4*a^3 + 39*a^2 + 126*a + 47]),K);
sage: E = EllipticCurve(K, [a^3 - 2*a^2 - 2*a + 1, -a^3 + 2*a^2 + 3*a - 2, a^2 - 2*a, 4*a^3 - 24*a^2 - 22*a - 8, 4*a^3 + 39*a^2 + 126*a + 47])
gp (2.8): E = ellinit([a^3 - 2*a^2 - 2*a + 1, -a^3 + 2*a^2 + 3*a - 2, a^2 - 2*a, 4*a^3 - 24*a^2 - 22*a - 8, 4*a^3 + 39*a^2 + 126*a + 47],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((49,a^{3} - a^{2} - 4 a + 1)\) = \( \left(-a^{3} + 3 a^{2} + a - 3\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 49 \) = \( 7^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((117649,a^{3} - 2 a^{2} - 2 a + 38182,a + 116652,a^{2} - 2 a + 66826)\) = \( \left(-a^{3} + 3 a^{2} + a - 3\right)^{6} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 117649 \) = \( 7^{6} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( 427641453408640 a^{3} + 331260963910720 a^{2} - 363800343820288 a - 154126073580480 \)
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: \(\Z/2\Z\)
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: $\left(2 a^{3} - 2 a^{2} - 6 a - \frac{7}{2} : \frac{7}{4} a^{3} - 4 a^{2} - \frac{9}{2} a - \frac{1}{4} : 1\right)$
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\((j)_{-}\)
\( \left(-a^{3} + 3 a^{2} + a - 3\right) \) \(7\) \(4\) \(I_{0}^*\) Additive \(-1\) \(2\) \(6\) \(0\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 49.4-a consists of curves linked by isogenies of degrees dividing 6.

Base change

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