Properties

Base field 6.6.434581.1
Label 6.6.434581.1-71.2-a4
Conductor \((71,2 a^{4} - 4 a^{3} - 6 a^{2} + 7 a + 2)\)
Conductor norm \( 71 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field 6.6.434581.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 5*x^3 + 4*x^2 - 2*x - 1)
 
gp: K = nfinit(a^6 - 2*a^5 - 4*a^4 + 5*a^3 + 4*a^2 - 2*a - 1);
 

Weierstrass equation

\( y^2 + a x y + \left(3 a^{5} - 7 a^{4} - 8 a^{3} + 15 a^{2} + a - 4\right) y = x^{3} + \left(-a^{5} + 2 a^{4} + 4 a^{3} - 4 a^{2} - 4 a + 2\right) x^{2} + \left(20 a^{5} - 51 a^{4} - 52 a^{3} + 120 a^{2} + 55 a - 84\right) x + 251 a^{5} - 630 a^{4} - 644 a^{3} + 1463 a^{2} + 269 a - 526 \)
magma: E := ChangeRing(EllipticCurve([a, -a^5 + 2*a^4 + 4*a^3 - 4*a^2 - 4*a + 2, 3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 4, 20*a^5 - 51*a^4 - 52*a^3 + 120*a^2 + 55*a - 84, 251*a^5 - 630*a^4 - 644*a^3 + 1463*a^2 + 269*a - 526]),K);
 
sage: E = EllipticCurve(K, [a, -a^5 + 2*a^4 + 4*a^3 - 4*a^2 - 4*a + 2, 3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 4, 20*a^5 - 51*a^4 - 52*a^3 + 120*a^2 + 55*a - 84, 251*a^5 - 630*a^4 - 644*a^3 + 1463*a^2 + 269*a - 526])
 
gp: E = ellinit([a, -a^5 + 2*a^4 + 4*a^3 - 4*a^2 - 4*a + 2, 3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 4, 20*a^5 - 51*a^4 - 52*a^3 + 120*a^2 + 55*a - 84, 251*a^5 - 630*a^4 - 644*a^3 + 1463*a^2 + 269*a - 526],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((71,2 a^{4} - 4 a^{3} - 6 a^{2} + 7 a + 2)\) = \( \left(a^{3} - 3 a\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 71 \) = \( 71 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((5041,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 4105,a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 4205,a + 3720,a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a + 2309,a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} + 2775)\) = \( \left(a^{3} - 3 a\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 5041 \) = \( 71^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{2427847058697256047703464284364}{5041} a^{5} - \frac{3213439376084993538620895804626}{5041} a^{4} - \frac{11885036707571320938161451911154}{5041} a^{3} + \frac{4099907686861892112740975364149}{5041} a^{2} + \frac{12484665412014987227895045256073}{5041} a + \frac{3589236914441632306388327741777}{5041} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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()
 

Regulator: not available

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

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(5 a^{5} - 12 a^{4} - 14 a^{3} + \frac{111}{4} a^{2} + 6 a - 8 : -\frac{1}{2} a^{5} + \frac{1}{2} a^{4} + \frac{21}{8} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a - \frac{1}{2} : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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\((j)_{-}\)
\( \left(a^{3} - 3 a\right) \) \(71\) \(2\) \(I_{2}\) Non-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
\(2\) 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 71.2-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.