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Properties

Label 6.6.1387029.1-49.1-d1
Base field 6.6.1387029.1
Conductor norm \( 49 \)
CM no
Base change yes
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.1387029.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
 
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-2a^{4}-4a^{3}+6a^{2}+3a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(2a^{5}-5a^{4}-7a^{3}+15a^{2}+5a-4\right){x}^{2}+\left(14a^{5}-35a^{4}-45a^{3}+102a^{2}+27a-41\right){x}+129a^{5}-326a^{4}-365a^{3}+977a^{2}+170a-433\)
sage: E = EllipticCurve([K([-2,3,6,-4,-2,1]),K([-4,5,15,-7,-5,2]),K([0,-4,0,1,0,0]),K([-41,27,102,-45,-35,14]),K([-433,170,977,-365,-326,129])])
 
gp: E = ellinit([Polrev([-2,3,6,-4,-2,1]),Polrev([-4,5,15,-7,-5,2]),Polrev([0,-4,0,1,0,0]),Polrev([-41,27,102,-45,-35,14]),Polrev([-433,170,977,-365,-326,129])], K);
 
magma: E := EllipticCurve([K![-2,3,6,-4,-2,1],K![-4,5,15,-7,-5,2],K![0,-4,0,1,0,0],K![-41,27,102,-45,-35,14],K![-433,170,977,-365,-326,129]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-2a^3-3a^2+4a+2)\) = \((a^2-2)^{2}\)
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: \((3a^4-6a^3-8a^2+11a-13)\) = \((a^2-2)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5764801 \) = \(7^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 861413641629 a^{4} - 1722827283258 a^{3} - 2337478594271 a^{2} + 3198892235900 a - 669598945129 \)
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: \( 70.894583931049616883588492304949261443 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 1.62530 \)
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\((j)_{-}\)
\((a^2-2)\) \(7\) \(3\) \(IV^{*}\) Additive \(1\) \(2\) \(8\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 49.1-d consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
3.3.257.1 3.3.257.1-49.2-b2
3.3.257.1 a curve with conductor norm 441 (not in the database)