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Properties

Label 6.6.485125.1-49.1-d1
Base field 6.6.485125.1
Conductor norm \( 49 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
 
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-a^{3}-3a^{2}+3a+2\right){x}{y}+\left(a^{5}-5a^{3}+4a\right){y}={x}^{3}+\left(-2a^{5}+3a^{4}+8a^{3}-10a^{2}-5a+4\right){x}^{2}+\left(-13a^{5}+21a^{4}+55a^{3}-79a^{2}-41a+43\right){x}-20a^{5}+30a^{4}+84a^{3}-118a^{2}-64a+68\)
sage: E = EllipticCurve([K([2,3,-3,-1,1,0]),K([4,-5,-10,8,3,-2]),K([0,4,0,-5,0,1]),K([43,-41,-79,55,21,-13]),K([68,-64,-118,84,30,-20])])
 
gp: E = ellinit([Polrev([2,3,-3,-1,1,0]),Polrev([4,-5,-10,8,3,-2]),Polrev([0,4,0,-5,0,1]),Polrev([43,-41,-79,55,21,-13]),Polrev([68,-64,-118,84,30,-20])], K);
 
magma: E := EllipticCurve([K![2,3,-3,-1,1,0],K![4,-5,-10,8,3,-2],K![0,4,0,-5,0,1],K![43,-41,-79,55,21,-13],K![68,-64,-118,84,30,-20]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) = \((-2a^5+3a^4+10a^3-12a^2-11a+6)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) = \((-2a^5+3a^4+10a^3-12a^2-11a+6)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 49 \) = \(49\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{6613630491}{7} a^{5} + \frac{8709281037}{7} a^{4} + \frac{32409828699}{7} a^{3} - \frac{30787577825}{7} a^{2} - \frac{34253730020}{7} a + \frac{9688739659}{7} \)
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 $\left(-a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 9 a + 7 : -5 a^{5} + 10 a^{4} + 21 a^{3} - 37 a^{2} - 14 a + 18 : 1\right)$
Height \(0.0082077579989152107749377594588408252987\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.0082077579989152107749377594588408252987 \)
Period: \( 36705.148891553005796020729552707625112 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.59523 \)
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\((j)_{-}\)
\((-2a^5+3a^4+10a^3-12a^2-11a+6)\) \(49\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 49.1-d consists of this curve only.

Base change

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

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