Properties

Label 5.5.38569.1-49.1-a2
Base field 5.5.38569.1
Conductor norm \( 49 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 5.5.38569.1

Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} + 4 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 0, -5, 0, 1]))
 
gp: K = nfinit(Polrev([-1, 4, 0, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 0, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{4}+a^{3}-9a^{2}-4a+5\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{4}-4a^{2}+2\right){x}^{2}+\left(-4a^{3}-10a^{2}-a+7\right){x}-11a^{4}-23a^{3}+5a^{2}+15a-1\)
sage: E = EllipticCurve([K([5,-4,-9,1,2]),K([2,0,-4,0,1]),K([1,-3,0,1,0]),K([7,-1,-10,-4,0]),K([-1,15,5,-23,-11])])
 
gp: E = ellinit([Polrev([5,-4,-9,1,2]),Polrev([2,0,-4,0,1]),Polrev([1,-3,0,1,0]),Polrev([7,-1,-10,-4,0]),Polrev([-1,15,5,-23,-11])], K);
 
magma: E := EllipticCurve([K![5,-4,-9,1,2],K![2,0,-4,0,1],K![1,-3,0,1,0],K![7,-1,-10,-4,0],K![-1,15,5,-23,-11]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4-a^3+4a^2+3a)\) = \((a^2-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((12a^4+17a^3-32a^2-84a-16)\) = \((a^2-2)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 282475249 \) = \(7^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1243808100}{2401} a^{4} + \frac{1002203176}{2401} a^{3} - \frac{5388495975}{2401} a^{2} - \frac{4276678300}{2401} a + \frac{1561357008}{2401} \)
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: $\left(-a^{4} - a^{3} + 4 a^{2} + 3 a - 3 : a^{4} - 4 a^{2} + a + 3 : 1\right)$
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: \( 616.80423543165618625486925763330402813 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.57035616 \)
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\((j)_{-}\)
\((a^2-2)\) \(7\) \(2\) \(I_{4}^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)

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 49.1-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.