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Properties

Base field 6.6.300125.1
Label 6.6.300125.1-71.4-a1
Conductor \((71,-2 a^{5} + 2 a^{4} + 13 a^{3} - 3 a^{2} - 8 a + 1)\)
Conductor norm \( 71 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field 6.6.300125.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 7*x^4 + 2*x^3 + 7*x^2 - 2*x - 1)
 
gp (2.8): K = nfinit(a^6 - a^5 - 7*a^4 + 2*a^3 + 7*a^2 - 2*a - 1);
 

Weierstrass equation

\( y^2 + \left(-4 a^{5} + a^{4} + 29 a^{3} + 14 a^{2} - 19 a - 6\right) x y + \left(-5 a^{5} + a^{4} + 37 a^{3} + 18 a^{2} - 27 a - 8\right) y = x^{3} + \left(11 a^{5} - 3 a^{4} - 79 a^{3} - 36 a^{2} + 52 a + 17\right) x^{2} + \left(-74 a^{5} + 22 a^{4} + 533 a^{3} + 231 a^{2} - 348 a - 101\right) x - 794 a^{5} + 224 a^{4} + 5719 a^{3} + 2523 a^{2} - 3738 a - 1101 \)
magma: E := ChangeRing(EllipticCurve([-4*a^5 + a^4 + 29*a^3 + 14*a^2 - 19*a - 6, 11*a^5 - 3*a^4 - 79*a^3 - 36*a^2 + 52*a + 17, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 27*a - 8, -74*a^5 + 22*a^4 + 533*a^3 + 231*a^2 - 348*a - 101, -794*a^5 + 224*a^4 + 5719*a^3 + 2523*a^2 - 3738*a - 1101]),K);
 
sage: E = EllipticCurve(K, [-4*a^5 + a^4 + 29*a^3 + 14*a^2 - 19*a - 6, 11*a^5 - 3*a^4 - 79*a^3 - 36*a^2 + 52*a + 17, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 27*a - 8, -74*a^5 + 22*a^4 + 533*a^3 + 231*a^2 - 348*a - 101, -794*a^5 + 224*a^4 + 5719*a^3 + 2523*a^2 - 3738*a - 1101])
 
gp (2.8): E = ellinit([-4*a^5 + a^4 + 29*a^3 + 14*a^2 - 19*a - 6, 11*a^5 - 3*a^4 - 79*a^3 - 36*a^2 + 52*a + 17, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 27*a - 8, -74*a^5 + 22*a^4 + 533*a^3 + 231*a^2 - 348*a - 101, -794*a^5 + 224*a^4 + 5719*a^3 + 2523*a^2 - 3738*a - 1101],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((71,-2 a^{5} + 2 a^{4} + 13 a^{3} - 3 a^{2} - 8 a + 1)\) = \( \left(6 a^{5} - 2 a^{4} - 43 a^{3} - 17 a^{2} + 29 a + 5\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 71 \) = \( 71 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((1804229351,-2 a^{5} + 15 a^{3} + 10 a^{2} - 10 a + 1541178947,3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 12 a + 911516894,-4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a + 683542689,a + 329597730,-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a + 1543737530)\) = \( \left(6 a^{5} - 2 a^{4} - 43 a^{3} - 17 a^{2} + 29 a + 5\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 1804229351 \) = \( 71^{5} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{704803031197900}{1804229351} a^{5} - \frac{2010453848133504}{1804229351} a^{4} - \frac{2574645568787336}{1804229351} a^{3} + \frac{8015459823309236}{1804229351} a^{2} - \frac{2156284877910108}{1804229351} a - \frac{1240401241894705}{1804229351} \)
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: $\left(14 a^{5} - 4 a^{4} - 101 a^{3} - 44 a^{2} + 66 a + 18 : 15 a^{5} - 4 a^{4} - 108 a^{3} - 48 a^{2} + 72 a + 21 : 1\right)$
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\((j)_{-}\)
\( \left(6 a^{5} - 2 a^{4} - 43 a^{3} - 17 a^{2} + 29 a + 5\right) \) \(71\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 71.4-a consists of curves linked by isogenies of degree 2.

Base change

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