↑

Properties

Base field 6.6.300125.1
Label 6.6.300125.1-71.3-a2
Conductor \((71,8 a^{5} - 2 a^{4} - 58 a^{3} - 27 a^{2} + 38 a + 10)\)
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 6.6.300125.1

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

\( y^2 + \left(3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 13 a + 4\right) x y + \left(-3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 6\right) y = x^{3} + \left(-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 11 a - 1\right) x^{2} + \left(-7 a^{5} - 2 a^{4} + 55 a^{3} + 44 a^{2} - 32 a - 25\right) x + 88 a^{5} - 195 a^{4} - 254 a^{3} + 186 a^{2} + 98 a - 25 \)
magma: E := ChangeRing(EllipticCurve([3*a^5 - a^4 - 21*a^3 - 9*a^2 + 13*a + 4, -2*a^5 + a^4 + 14*a^3 + 4*a^2 - 11*a - 1, -3*a^5 + 23*a^3 + 14*a^2 - 17*a - 6, -7*a^5 - 2*a^4 + 55*a^3 + 44*a^2 - 32*a - 25, 88*a^5 - 195*a^4 - 254*a^3 + 186*a^2 + 98*a - 25]),K);
 
sage: E = EllipticCurve(K, [3*a^5 - a^4 - 21*a^3 - 9*a^2 + 13*a + 4, -2*a^5 + a^4 + 14*a^3 + 4*a^2 - 11*a - 1, -3*a^5 + 23*a^3 + 14*a^2 - 17*a - 6, -7*a^5 - 2*a^4 + 55*a^3 + 44*a^2 - 32*a - 25, 88*a^5 - 195*a^4 - 254*a^3 + 186*a^2 + 98*a - 25])
 
gp (2.8): E = ellinit([3*a^5 - a^4 - 21*a^3 - 9*a^2 + 13*a + 4, -2*a^5 + a^4 + 14*a^3 + 4*a^2 - 11*a - 1, -3*a^5 + 23*a^3 + 14*a^2 - 17*a - 6, -7*a^5 - 2*a^4 + 55*a^3 + 44*a^2 - 32*a - 25, 88*a^5 - 195*a^4 - 254*a^3 + 186*a^2 + 98*a - 25],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((71,8 a^{5} - 2 a^{4} - 58 a^{3} - 27 a^{2} + 38 a + 10)\) = \( \left(-a^{5} + a^{4} + 7 a^{3} - 2 a^{2} - 7 a\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 71 \) = \( 71 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((3255243551009881201,-2 a^{5} + 15 a^{3} + 10 a^{2} - 10 a + 965425644278867757,3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 12 a + 1553260087533172811,-4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a + 860602312777624240,a + 1168525495096135975,-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a + 2269114039968732632)\) = \( \left(-a^{5} + a^{4} + 7 a^{3} - 2 a^{2} - 7 a\right)^{10} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 3255243551009881201 \) = \( 71^{10} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{7245137537592799459668364}{3255243551009881201} a^{5} - \frac{17175864433733831866928686}{3255243551009881201} a^{4} - \frac{29002126063391338539879632}{3255243551009881201} a^{3} + \frac{59053137968044220795988828}{3255243551009881201} a^{2} - \frac{333346305063766203121436}{45848500718449031} a + \frac{536537380242287614298139}{3255243551009881201} \)
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: $\left(\frac{7}{2} a^{5} - a^{4} - \frac{105}{4} a^{3} - \frac{33}{4} a^{2} + \frac{37}{2} a + \frac{9}{4} : -a^{5} + \frac{7}{4} a^{4} + \frac{33}{8} a^{3} - 2 a^{2} - \frac{5}{4} a + \frac{1}{2} : 1\right)$
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\((j)_{-}\)
\( \left(-a^{5} + a^{4} + 7 a^{3} - 2 a^{2} - 7 a\right) \) \(71\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)

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 71.3-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.