Properties

Label 5.5.24217.1-97.1-a1
Base field 5.5.24217.1
Conductor norm \( 97 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-a^3-5a^2+5a+4)\) = \((a^4-a^3-5a^2+5a+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 97 \) = \(97\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^4+a^3+5a^2-5a-4)\) = \((a^4-a^3-5a^2+5a+4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 97 \) = \(97\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{761279}{97} a^{4} - \frac{433426}{97} a^{3} + \frac{3272354}{97} a^{2} + \frac{2641857}{97} a + \frac{440084}{97} \)
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: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 a^{4} - a^{3} - 9 a^{2} + 2 a + 4 : a^{4} - 6 a^{2} - a + 5 : 1\right)$
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: \( 2396.4286055045978563557210415510866650 \)
Tamagawa product: \( 1 \)
Torsion order: \(4\)
Leading coefficient: \( 0.962463662 \)
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\((j)_{-}\)
\((a^4-a^3-5a^2+5a+4)\) \(97\) \(1\) \(I_{1}\) 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 and 4.
Its isogeny class 97.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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