Properties

Label 5.5.65657.1-45.1-b2
Base field 5.5.65657.1
Conductor norm \( 45 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

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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 \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-7a-4\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}^{2}+\left(330a^{4}+154a^{3}-1529a^{2}-1577a-314\right){x}+9329a^{4}+3425a^{3}-41530a^{2}-38029a-6558\)
sage: E = EllipticCurve([K([0,1,0,0,0]),K([3,7,-4,-2,1]),K([-4,-7,5,2,-1]),K([-314,-1577,-1529,154,330]),K([-6558,-38029,-41530,3425,9329])])
 
gp: E = ellinit([Polrev([0,1,0,0,0]),Polrev([3,7,-4,-2,1]),Polrev([-4,-7,5,2,-1]),Polrev([-314,-1577,-1529,154,330]),Polrev([-6558,-38029,-41530,3425,9329])], K);
 
magma: E := EllipticCurve([K![0,1,0,0,0],K![3,7,-4,-2,1],K![-4,-7,5,2,-1],K![-314,-1577,-1529,154,330],K![-6558,-38029,-41530,3425,9329]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-4a)\) = \((-a^4+a^3+4a^2-2a-2)^{2}\cdot(-a^2+a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 45 \) = \(3^{2}\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^4+5a^3+7a^2-12a-7)\) = \((-a^4+a^3+4a^2-2a-2)^{6}\cdot(-a^2+a+2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -3645 \) = \(-3^{6}\cdot5\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-11 a^{4} - a^{3} + 49 a^{2} + 34 a + 8 : 83 a^{4} + 42 a^{3} - 378 a^{2} - 394 a - 70 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 134.14023065512017269522700963416277868 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(3\)
Leading coefficient: \( 1.45417284 \)
Analytic order of Ш: \( 25 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((-a^4+a^3+4a^2-2a-2)\) \(3\) \(1\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((-a^2+a+2)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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 elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.