Properties

Label 5.5.65657.1-9.1-b2
Base field 5.5.65657.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 5.5.65657.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-11a-10\right){x}^{2}+\left(17a^{4}+247a^{3}-506a^{2}-251a-16\right){x}+3325a^{4}-6557a^{3}-5396a^{2}+5378a+1457\)
sage: E = EllipticCurve([K([-1,-4,0,1,0]),K([-10,-11,13,4,-3]),K([-2,0,1,0,0]),K([-16,-251,-506,247,17]),K([1457,5378,-5396,-6557,3325])])
 
gp: E = ellinit([Polrev([-1,-4,0,1,0]),Polrev([-10,-11,13,4,-3]),Polrev([-2,0,1,0,0]),Polrev([-16,-251,-506,247,17]),Polrev([1457,5378,-5396,-6557,3325])], K);
 
magma: E := EllipticCurve([K![-1,-4,0,1,0],K![-10,-11,13,4,-3],K![-2,0,1,0,0],K![-16,-251,-506,247,17],K![1457,5378,-5396,-6557,3325]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-2a^3-4a^2+5a+3)\) = \((-a^4+a^3+4a^2-2a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3a^4+2a^3+15a^2-6a-8)\) = \((-a^4+a^3+4a^2-2a-2)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -6561 \) = \(-3^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2521498274075249905348265612}{9} a^{4} - \frac{1488610650200171884252903124}{3} a^{3} - \frac{9163877093686100970416615593}{9} a^{2} + 1345476142192101801059271874 a + \frac{3269990960361707860438009855}{9} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 5.3659745798086978234956901075095161344 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 1.04707574 \)
Analytic order of Ш: \( 25 \) (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\((j)_{-}\)
\((-a^4+a^3+4a^2-2a-2)\) \(3\) \(2\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 9.1-b consists of curves linked by isogenies of degree 5.

Base change

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

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