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Properties

Base field 6.6.300125.1
Label 6.6.300125.1-71.1-a2
Conductor \((71,-8 a^{5} + a^{4} + 58 a^{3} + 34 a^{2} - 34 a - 16)\)
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

\( y^2 + \left(-6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 28 a - 7\right) x y + \left(-a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 2\right) y = x^{3} + \left(-7 a^{5} + 2 a^{4} + 50 a^{3} + 22 a^{2} - 30 a - 10\right) x^{2} + \left(2 a^{5} + 7 a^{4} - 24 a^{3} - 51 a^{2} + 29 a + 8\right) x - 327 a^{5} + 677 a^{4} + 1538 a^{3} - 2267 a^{2} + 286 a + 277 \)
magma: E := ChangeRing(EllipticCurve([-6*a^5 + 2*a^4 + 43*a^3 + 17*a^2 - 28*a - 7, -7*a^5 + 2*a^4 + 50*a^3 + 22*a^2 - 30*a - 10, -a^5 + 8*a^3 + 5*a^2 - 7*a - 2, 2*a^5 + 7*a^4 - 24*a^3 - 51*a^2 + 29*a + 8, -327*a^5 + 677*a^4 + 1538*a^3 - 2267*a^2 + 286*a + 277]),K);
 
sage: E = EllipticCurve(K, [-6*a^5 + 2*a^4 + 43*a^3 + 17*a^2 - 28*a - 7, -7*a^5 + 2*a^4 + 50*a^3 + 22*a^2 - 30*a - 10, -a^5 + 8*a^3 + 5*a^2 - 7*a - 2, 2*a^5 + 7*a^4 - 24*a^3 - 51*a^2 + 29*a + 8, -327*a^5 + 677*a^4 + 1538*a^3 - 2267*a^2 + 286*a + 277])
 
gp (2.8): E = ellinit([-6*a^5 + 2*a^4 + 43*a^3 + 17*a^2 - 28*a - 7, -7*a^5 + 2*a^4 + 50*a^3 + 22*a^2 - 30*a - 10, -a^5 + 8*a^3 + 5*a^2 - 7*a - 2, 2*a^5 + 7*a^4 - 24*a^3 - 51*a^2 + 29*a + 8, -327*a^5 + 677*a^4 + 1538*a^3 - 2267*a^2 + 286*a + 277],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((71,-8 a^{5} + a^{4} + 58 a^{3} + 34 a^{2} - 34 a - 16)\) = \( \left(4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 18 a + 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 71 \) = \( 71 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((3255243551009881201,-2 a^{5} + 15 a^{3} + 10 a^{2} - 10 a + 1701983463476708389,3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 12 a + 2667409107755576159,-4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a + 2394641238232256952,a + 398295067786843521,-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a + 77572652553743889)\) = \( \left(4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 18 a + 3\right)^{10} \)
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)
 
\(j\) = \( \frac{182367117100063826098201858}{3255243551009881201} a^{5} - \frac{50495473559193738950995130}{3255243551009881201} a^{4} - \frac{1314604738469846691446620746}{3255243551009881201} a^{3} - \frac{582379993113800706094818270}{3255243551009881201} a^{2} + \frac{859358590365859284077183254}{3255243551009881201} a + \frac{253657973068162907887516563}{3255243551009881201} \)
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: $\left(\frac{5}{4} a^{5} - 3 a^{4} - 5 a^{3} + \frac{41}{4} a^{2} - \frac{7}{2} a + \frac{1}{2} : \frac{31}{8} a^{5} - \frac{19}{8} a^{4} - \frac{53}{2} a^{3} - \frac{39}{8} a^{2} + 16 a + \frac{23}{8} : 1\right)$
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\((j)_{-}\)
\( \left(4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 18 a + 3\right) \) \(71\) \(2\) \(I_{10}\) 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 71.1-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.