Properties

Base field \(\Q(\zeta_{7})^+\)
Label 3.3.49.1-71.2-a2
Conductor \((71,-3 a^{2} + 4 a + 5)\)
Conductor norm \( 71 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field \(\Q(\zeta_{7})^+\)

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

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

Weierstrass equation

\( y^2 + x y + \left(a^{2} + a - 2\right) y = x^{3} + \left(-a^{2} + a + 1\right) x^{2} + \left(5 a^{2} + 4 a - 50\right) x + 92 a^{2} - 32 a - 290 \)
magma: E := ChangeRing(EllipticCurve([1, -a^2 + a + 1, a^2 + a - 2, 5*a^2 + 4*a - 50, 92*a^2 - 32*a - 290]),K);
 
sage: E = EllipticCurve(K, [1, -a^2 + a + 1, a^2 + a - 2, 5*a^2 + 4*a - 50, 92*a^2 - 32*a - 290])
 
gp: E = ellinit([1, -a^2 + a + 1, a^2 + a - 2, 5*a^2 + 4*a - 50, 92*a^2 - 32*a - 290],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((71,-3 a^{2} + 4 a + 5)\) = \( \left(-3 a^{2} + 4 a + 5\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 71 \) = \( 71 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((1804229351,a + 263050399,a^{2} + 1174567288)\) = \( \left(-3 a^{2} + 4 a + 5\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 1804229351 \) = \( 71^{5} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{539799973605204231}{1804229351} a^{2} - \frac{925759129064909928}{1804229351} a + \frac{305086529379437264}{1804229351} \)
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: $\left(-5 a^{2} + 3 a + 8 : 2 a^{2} - 2 a - 3 : 1\right)$
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\((j)_{-}\)
\( \left(-3 a^{2} + 4 a + 5\right) \) \(71\) \(5\) \(I_{5}\) 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
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 71.2-a consists of curves linked by isogenies of degrees dividing 10.

Base change

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