Properties

Label 6.6.592661.1-73.2-b1
Base field 6.6.592661.1
Conductor norm \( 73 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 6.6.592661.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-a^{4}-9a^{3}+4a^{2}+5a-3\right){x}{y}+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+5a-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-15a^{5}+33a^{4}+46a^{3}-126a^{2}+28a+25\right){x}-62a^{5}+155a^{4}+190a^{3}-590a^{2}+117a+98\)
sage: E = EllipticCurve([K([-3,5,4,-9,-1,2]),K([3,0,-1,0,0,0]),K([-4,5,5,-9,-1,2]),K([25,28,-126,46,33,-15]),K([98,117,-590,190,155,-62])])
 
gp: E = ellinit([Polrev([-3,5,4,-9,-1,2]),Polrev([3,0,-1,0,0,0]),Polrev([-4,5,5,-9,-1,2]),Polrev([25,28,-126,46,33,-15]),Polrev([98,117,-590,190,155,-62])], K);
 
magma: E := EllipticCurve([K![-3,5,4,-9,-1,2],K![3,0,-1,0,0,0],K![-4,5,5,-9,-1,2],K![25,28,-126,46,33,-15],K![98,117,-590,190,155,-62]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^5+2a^4+5a^3-8a^2-4a+3)\) = \((-a^5+2a^4+5a^3-8a^2-4a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 73 \) = \(73\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-54a^5+301a^4-5a^3-1177a^2+610a-263)\) = \((-a^5+2a^4+5a^3-8a^2-4a+3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 58871586708267913 \) = \(73^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{819409026299375534395416}{58871586708267913} a^{5} - \frac{170251602571138272825926}{58871586708267913} a^{4} + \frac{3968508698135623357582134}{58871586708267913} a^{3} + \frac{1769334738717973761105523}{58871586708267913} a^{2} - \frac{1771210861015578326012193}{58871586708267913} a - \frac{587323218030274014006283}{58871586708267913} \)
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: \(1\)
Generator $\left(-9 a^{5} + 7 a^{4} + 38 a^{3} - 27 a^{2} - 15 a + 17 : -69 a^{5} + 34 a^{4} + 307 a^{3} - 124 a^{2} - 161 a + 97 : 1\right)$
Height \(0.13629071148651834659514730526851115249\)
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: $\left(a^{5} + a^{4} - 6 a^{3} - 4 a^{2} + 7 a - 1 : 4 a^{5} - a^{4} - 18 a^{3} + 2 a^{2} + 10 a - 2 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.13629071148651834659514730526851115249 \)
Period: \( 122.21723548175440316502011731765875640 \)
Tamagawa product: \( 9 \)
Torsion order: \(2\)
Leading coefficient: \( 2.62889 \)
Analytic order of Ш: \( 9 \) (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\((j)_{-}\)
\((-a^5+2a^4+5a^3-8a^2-4a+3)\) \(73\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)

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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 73.2-b consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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