Properties

Base field 4.4.4205.1
Label 4.4.4205.1-49.4-c1
Conductor \((49,2 a^{3} - 3 a^{2} - 7 a)\)
Conductor norm \( 49 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Base field 4.4.4205.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} - x + 1 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 5*x^2 - x + 1)
 
gp: K = nfinit(a^4 - a^3 - 5*a^2 - a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);
 

Weierstrass equation

\( y^2 + \left(a^{3} - a^{2} - 4 a - 1\right) x y + \left(a + 1\right) y = x^{3} + \left(-2 a^{3} + 3 a^{2} + 8 a + 1\right) x^{2} + \left(66 a^{3} - 37 a^{2} - 364 a - 233\right) x - 457 a^{3} + 318 a^{2} + 2384 a + 1238 \)
sage: E = EllipticCurve(K, [a^3 - a^2 - 4*a - 1, -2*a^3 + 3*a^2 + 8*a + 1, a + 1, 66*a^3 - 37*a^2 - 364*a - 233, -457*a^3 + 318*a^2 + 2384*a + 1238])
 
gp: E = ellinit([a^3 - a^2 - 4*a - 1, -2*a^3 + 3*a^2 + 8*a + 1, a + 1, 66*a^3 - 37*a^2 - 364*a - 233, -457*a^3 + 318*a^2 + 2384*a + 1238],K)
 
magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 4*a - 1, -2*a^3 + 3*a^2 + 8*a + 1, a + 1, 66*a^3 - 37*a^2 - 364*a - 233, -457*a^3 + 318*a^2 + 2384*a + 1238]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((49,2 a^{3} - 3 a^{2} - 7 a)\) = \( \left(-a^{3} + 2 a^{2} + 3 a - 3\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 49 \) = \( 7^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((5764801,a + 2899171,a^{3} - a^{2} - 5 a + 3123108,-a^{3} + 2 a^{2} + 3 a + 5274931)\) = \( \left(-a^{3} + 2 a^{2} + 3 a - 3\right)^{8} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 5764801 \) = \( 7^{8} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -156521478494 a^{3} + 257550722470 a^{2} + 616417035615 a - 241680788762 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

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

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a^{3} + 2 a^{2} + 3 a - 3\right) \) \(7\) \(1\) \(IV^*\) Additive \(1\) \(2\) \(8\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.6.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 49.4-c consists of curves linked by isogenies of degree 7.

Base change

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