Properties

Base field 3.3.49.1
Label 3.3.49.1-41.1-a1
Conductor \((41,a^{2} + 2 a - 4)\)
Conductor norm \( 41 \)
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

\( y^2 + x y + y = x^{3} + \left(-a^{2} + 3\right) x^{2} + \left(99 a^{2} - 10 a - 348\right) x + 952 a^{2} - 216 a - 2798 \)
magma: E := ChangeRing(EllipticCurve([1, -a^2 + 3, 1, 99*a^2 - 10*a - 348, 952*a^2 - 216*a - 2798]),K);
sage: E = EllipticCurve(K, [1, -a^2 + 3, 1, 99*a^2 - 10*a - 348, 952*a^2 - 216*a - 2798])
gp (2.8): E = ellinit([1, -a^2 + 3, 1, 99*a^2 - 10*a - 348, 952*a^2 - 216*a - 2798],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((41,a^{2} + 2 a - 4)\) = \( \left(a^{2} + 2 a - 4\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 41 \) = \( 41 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((13422659310152401,a + 2146762310336197,a^{2} + 11763256385457480)\) = \( \left(a^{2} + 2 a - 4\right)^{10} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 13422659310152401 \) = \( 41^{10} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{182915726357803972950650}{13422659310152401} a^{2} - \frac{357571850055303381213985}{13422659310152401} a + \frac{50482569444763032743584}{13422659310152401} \)
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: $\left(-11 a^{2} + 8 a + \frac{75}{4} : \frac{11}{2} a^{2} - 4 a - \frac{79}{8} : 1\right)$
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\((j)_{-}\)
\( \left(a^{2} + 2 a - 4\right) \) 41 \(10\) \( I_{10} \) 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 41.1-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.