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Properties

Label 5.5.24217.1-85.1-b2
Base field 5.5.24217.1
Conductor norm \( 85 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

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Base field 5.5.24217.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
 
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(-2a^{4}+a^{3}+10a^{2}-2a-5\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(3a^{4}-a^{3}-14a^{2}+3a+6\right){x}^{2}+\left(4a^{4}-19a^{2}-3a+9\right){x}+4a^{4}-a^{3}-19a^{2}+a+8\)
sage: E = EllipticCurve([K([-5,-2,10,1,-2]),K([6,3,-14,-1,3]),K([-1,1,1,0,0]),K([9,-3,-19,0,4]),K([8,1,-19,-1,4])])
 
gp: E = ellinit([Polrev([-5,-2,10,1,-2]),Polrev([6,3,-14,-1,3]),Polrev([-1,1,1,0,0]),Polrev([9,-3,-19,0,4]),Polrev([8,1,-19,-1,4])], K);
 
magma: E := EllipticCurve([K![-5,-2,10,1,-2],K![6,3,-14,-1,3],K![-1,1,1,0,0],K![9,-3,-19,0,4],K![8,1,-19,-1,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a^4+a^3+14a^2-3a-5)\) = \((2a^4-a^3-9a^2+2a+3)\cdot(a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 85 \) = \(5\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3a^4-a^3-15a^2+6a+8)\) = \((2a^4-a^3-9a^2+2a+3)^{3}\cdot(a^2-2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -2125 \) = \(-5^{3}\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{7625258}{2125} a^{4} + \frac{7955226}{2125} a^{3} - \frac{33370843}{2125} a^{2} - \frac{2334587}{125} a - \frac{9844789}{2125} \)
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: $\left(-2 a^{4} + 9 a^{2} + a - 2 : a^{4} + a^{3} - 4 a^{2} - 5 a - 2 : 1\right)$
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: \( 640.39130113546841417655050909656320961 \)
Tamagawa product: \( 3 \)  =  \(3\cdot1\)
Torsion order: \(3\)
Leading coefficient: \( 1.37171535 \)
Analytic order of Ш: \( 1 \) (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\((j)_{-}\)
\((2a^4-a^3-9a^2+2a+3)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a^2-2)\) \(17\) \(1\) \(I_{1}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 85.1-b consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

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