Properties

Label 5.5.24217.1-97.1-a2
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

\({y}^2+\left(-a^{4}+a^{3}+5a^{2}-3a-2\right){x}{y}+\left(a^{4}-4a^{2}+a\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(13a^{4}+22a^{3}-51a^{2}-110a-45\right){x}+72a^{4}+75a^{3}-298a^{2}-393a-125\)
sage: E = EllipticCurve([K([-2,-3,5,1,-1]),K([0,-4,-1,1,0]),K([0,1,-4,0,1]),K([-45,-110,-51,22,13]),K([-125,-393,-298,75,72])])
 
gp: E = ellinit([Polrev([-2,-3,5,1,-1]),Polrev([0,-4,-1,1,0]),Polrev([0,1,-4,0,1]),Polrev([-45,-110,-51,22,13]),Polrev([-125,-393,-298,75,72])], K);
 
magma: E := EllipticCurve([K![-2,-3,5,1,-1],K![0,-4,-1,1,0],K![0,1,-4,0,1],K![-45,-110,-51,22,13],K![-125,-393,-298,75,72]]);
 

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: \((-3a^4+13a^2+4a-9)\) = \((a^4-a^3-5a^2+5a+4)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 9409 \) = \(97^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2699246979785}{9409} a^{4} - \frac{2388620148409}{9409} a^{3} + \frac{11413457143722}{9409} a^{2} + \frac{12792778212781}{9409} a + \frac{3119261293489}{9409} \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(4 a^{4} - a^{3} - 18 a^{2} + 6 : -a^{4} + a^{3} + 4 a^{2} - 4 a - 1 : 1\right)$ $\left(\frac{1}{2} a^{4} + \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{11}{4} a - \frac{13}{4} : -\frac{3}{2} a^{4} + \frac{9}{8} a^{3} + \frac{53}{8} a^{2} - \frac{9}{2} a - \frac{19}{8} : 1\right)$
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: \( 299.55357568807473204446513019388583313 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 0.962463662 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((a^4-a^3-5a^2+5a+4)\) \(97\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
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.