Properties

Base field 4.4.2624.1
Label 4.4.2624.1-49.4-a1
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

\( y^2 + \left(a^{3} - a^{2} - 3 a\right) x y + a y = x^{3} + \left(a^{3} - a^{2} - 4 a + 1\right) x^{2} + \left(38 a^{3} - 41 a^{2} - 159 a - 66\right) x - 241 a^{3} + 276 a^{2} + 974 a + 319 \)
magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 3*a, a^3 - a^2 - 4*a + 1, a, 38*a^3 - 41*a^2 - 159*a - 66, -241*a^3 + 276*a^2 + 974*a + 319]),K);
 
sage: E = EllipticCurve(K, [a^3 - a^2 - 3*a, a^3 - a^2 - 4*a + 1, a, 38*a^3 - 41*a^2 - 159*a - 66, -241*a^3 + 276*a^2 + 974*a + 319])
 
gp (2.8): E = ellinit([a^3 - a^2 - 3*a, a^3 - a^2 - 4*a + 1, a, 38*a^3 - 41*a^2 - 159*a - 66, -241*a^3 + 276*a^2 + 974*a + 319],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)
 
\(j\) = \( 3292211757861632 a^{3} - 7770967386451264 a^{2} - 7075906078720256 a + 9134646147509504 \)
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: $\left(-2 a^{3} + \frac{3}{2} a^{2} + 6 a + \frac{7}{2} : -\frac{5}{4} a^{3} + 4 a^{2} + 8 a + \frac{7}{4} : 1\right)$
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\((j)_{-}\)
\( \left(-a^{3} + 3 a^{2} + a - 3\right) \) \(7\) \(2\) \(I_{0}^*\) 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.