Properties

Base field 5.5.65657.1
Label 5.5.65657.1-45.1-b2
Conductor \((45,-a^{3} + 4 a)\)
Conductor norm \( 45 \)
CM no
base-change no
Q-curve no
Torsion order \( 3 \)
Rank not available

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Base field 5.5.65657.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)
 
gp (2.8): K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);
 

Weierstrass equation

\( y^2 + a x y + \left(-a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 4\right) y = x^{3} + \left(a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\right) x^{2} + \left(330 a^{4} + 154 a^{3} - 1529 a^{2} - 1577 a - 314\right) x + 9329 a^{4} + 3425 a^{3} - 41530 a^{2} - 38029 a - 6558 \)
magma: E := ChangeRing(EllipticCurve([a, a^4 - 2*a^3 - 4*a^2 + 7*a + 3, -a^4 + 2*a^3 + 5*a^2 - 7*a - 4, 330*a^4 + 154*a^3 - 1529*a^2 - 1577*a - 314, 9329*a^4 + 3425*a^3 - 41530*a^2 - 38029*a - 6558]),K);
 
sage: E = EllipticCurve(K, [a, a^4 - 2*a^3 - 4*a^2 + 7*a + 3, -a^4 + 2*a^3 + 5*a^2 - 7*a - 4, 330*a^4 + 154*a^3 - 1529*a^2 - 1577*a - 314, 9329*a^4 + 3425*a^3 - 41530*a^2 - 38029*a - 6558])
 
gp (2.8): E = ellinit([a, a^4 - 2*a^3 - 4*a^2 + 7*a + 3, -a^4 + 2*a^3 + 5*a^2 - 7*a - 4, 330*a^4 + 154*a^3 - 1529*a^2 - 1577*a - 314, 9329*a^4 + 3425*a^3 - 41530*a^2 - 38029*a - 6558],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((45,-a^{3} + 4 a)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \cdot \left(-a^{2} + a + 2\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 45 \) = \( 3^{2} \cdot 5 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((3645,a + 1073,a^{4} - a^{3} - 4 a^{2} + 2 a + 1634,-a^{4} + a^{3} + 5 a^{2} - 3 a + 1429,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 3046)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{6} \cdot \left(-a^{2} + a + 2\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 3645 \) = \( 3^{6} \cdot 5 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \)
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: \(\Z/3\Z\)
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: $\left(-11 a^{4} - a^{3} + 49 a^{2} + 34 a + 8 : 83 a^{4} + 42 a^{3} - 378 a^{2} - 394 a - 70 : 1\right)$
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\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(1\) \(I_{0}^*\) Additive \(-1\) \(2\) \(6\) \(0\)
\( \left(-a^{2} + a + 2\right) \) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(3\) 3B.1.1
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 45.1-b consists of curves linked by isogenies of degrees dividing 15.

Base change

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