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Properties

Label 5.5.24217.1-85.2-a1
Base field 5.5.24217.1
Conductor norm \( 85 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+\left(2a^{4}-a^{3}-9a^{2}+4a+4\right){x}{y}={x}^{3}+\left(3a^{4}-2a^{3}-14a^{2}+5a+7\right){x}^{2}+\left(2a^{4}-a^{3}-9a^{2}+3a+6\right){x}\)
sage: E = EllipticCurve([K([4,4,-9,-1,2]),K([7,5,-14,-2,3]),K([0,0,0,0,0]),K([6,3,-9,-1,2]),K([0,0,0,0,0])])
 
gp: E = ellinit([Polrev([4,4,-9,-1,2]),Polrev([7,5,-14,-2,3]),Polrev([0,0,0,0,0]),Polrev([6,3,-9,-1,2]),Polrev([0,0,0,0,0])], K);
 
magma: E := EllipticCurve([K![4,4,-9,-1,2],K![7,5,-14,-2,3],K![0,0,0,0,0],K![6,3,-9,-1,2],K![0,0,0,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-5a^2-2a+3)\) = \((2a^4-a^3-9a^2+2a+3)\cdot(a^2-a-3)\)
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: \((-4a^4+3a^3+18a^2-9a-9)\) = \((2a^4-a^3-9a^2+2a+3)^{2}\cdot(a^2-a-3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 425 \) = \(5^{2}\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{62677552}{425} a^{4} + \frac{90221181}{425} a^{3} + \frac{148784492}{425} a^{2} - \frac{70726799}{425} a - \frac{76071034}{425} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(0 : 0 : 1\right)$
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: \( 346.83830448613195089223978004442825240 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.11438918 \)
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\((j)_{-}\)
\((2a^4-a^3-9a^2+2a+3)\) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a^2-a-3)\) \(17\) \(1\) \(I_{1}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 85.2-a consists of curves linked by isogenies of degree 2.

Base change

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

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