Properties

Base field 3.3.49.1
Label 3.3.49.1-71.1-a1
Conductor \((71,4 a^{2} - 3 a - 5)\)
Conductor norm \( 71 \)
CM no
base-change no
Q-curve not determined
Torsion order \( 2 \)
Rank not available

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Base field 3.3.49.1

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 (2.8): K = nfinit(a^3 - a^2 - 2*a + 1);

Weierstrass equation

\( y^2 + x y + \left(a + 1\right) y = x^{3} - a x^{2} + \left(-190 a^{2} + 410 a - 171\right) x - 2384 a^{2} + 5309 a - 1936 \)
magma: E := ChangeRing(EllipticCurve([1, -a, a + 1, -190*a^2 + 410*a - 171, -2384*a^2 + 5309*a - 1936]),K);
sage: E = EllipticCurve(K, [1, -a, a + 1, -190*a^2 + 410*a - 171, -2384*a^2 + 5309*a - 1936])
gp (2.8): E = ellinit([1, -a, a + 1, -190*a^2 + 410*a - 171, -2384*a^2 + 5309*a - 1936],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((71,4 a^{2} - 3 a - 5)\) = \( \left(4 a^{2} - 3 a - 5\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 71 \) = \( 71 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((3255243551009881201,a + 1553260087533172807,a^{2} + 965425644278867760)\) = \( \left(4 a^{2} - 3 a - 5\right)^{10} \)
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)
\(j\) = \( \frac{808193592087701284035551}{3255243551009881201} a^{2} - \frac{1936929577866441496465968}{3255243551009881201} a + \frac{752245811606895893913582}{3255243551009881201} \)
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 rank and generators

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: $\left(-4 a^{2} + 11 a + \frac{7}{4} : 2 a^{2} - 6 a - \frac{11}{8} : 1\right)$
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(4 a^{2} - 3 a - 5\right) \) 71 \(10\) \( I_{10} \) Split multiplicative 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
\(5\) 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 71.1-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 has not yet been determined whether or not it is a \(\Q\)-curve.