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Properties

Label 6.6.592661.1-49.2-d1
Base field 6.6.592661.1
Conductor norm \( 49 \)
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

\({y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-3\right){x}{y}+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+5a-5\right){y}={x}^{3}+\left(-a^{5}+5a^{3}+a^{2}-3a-2\right){x}^{2}+\left(27a^{5}-81a^{4}-85a^{3}+376a^{2}-22a-297\right){x}+242a^{5}-536a^{4}-965a^{3}+2432a^{2}+294a-1763\)
sage: E = EllipticCurve([K([-3,2,1,-4,0,1]),K([-2,-3,1,5,0,-1]),K([-5,5,5,-9,-1,2]),K([-297,-22,376,-85,-81,27]),K([-1763,294,2432,-965,-536,242])])
 
gp: E = ellinit([Polrev([-3,2,1,-4,0,1]),Polrev([-2,-3,1,5,0,-1]),Polrev([-5,5,5,-9,-1,2]),Polrev([-297,-22,376,-85,-81,27]),Polrev([-1763,294,2432,-965,-536,242])], K);
 
magma: E := EllipticCurve([K![-3,2,1,-4,0,1],K![-2,-3,1,5,0,-1],K![-5,5,5,-9,-1,2],K![-297,-22,376,-85,-81,27],K![-1763,294,2432,-965,-536,242]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-a^4-4a^3+4a^2+3a-1)\) = \((a^4-4a^2+a+2)^{2}\)
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: \((64a^5+54a^4-308a^3-325a^2+364a+300)\) = \((a^4-4a^2+a+2)^{16}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -33232930569601 \) = \(-7^{16}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{6378555847697446312}{282475249} a^{5} - \frac{2474411833285999287}{282475249} a^{4} + \frac{28471855515534194816}{282475249} a^{3} + \frac{13996124332632398039}{282475249} a^{2} - \frac{12521297730192365783}{282475249} a - \frac{4603543349670581566}{282475249} \)
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 $\left(34 a^{5} - 3 a^{4} - 183 a^{3} - 22 a^{2} + 199 a + 89 : 597 a^{5} - 226 a^{4} - 3104 a^{3} + 445 a^{2} + 3160 a + 814 : 1\right)$
Height \(0.14099245236947219113669285556434002091\)
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: $\left(\frac{3}{2} a^{5} - 3 a^{4} - 6 a^{3} + 14 a^{2} + \frac{3}{4} a - \frac{43}{4} : \frac{11}{4} a^{5} - \frac{39}{8} a^{4} - \frac{89}{8} a^{3} + \frac{85}{4} a^{2} + \frac{33}{8} a - \frac{57}{4} : 1\right)$
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: \( 0.14099245236947219113669285556434002091 \)
Period: \( 2368.4838185994092846599597334413058715 \)
Tamagawa product: \( 4 \)
Torsion order: \(2\)
Leading coefficient: \( 2.60264 \)
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\((j)_{-}\)
\((a^4-4a^2+a+2)\) \(7\) \(4\) \(I_{10}^{*}\) Additive \(-1\) \(2\) \(16\) \(10\)

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 49.2-d 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.