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Properties

Label 6.6.485125.1-49.1-f1
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

\({y}^2+\left(a+1\right){x}{y}+\left(2a^{5}-2a^{4}-9a^{3}+7a^{2}+8a-3\right){y}={x}^{3}+\left(-3a^{5}+4a^{4}+13a^{3}-14a^{2}-9a+5\right){x}^{2}+\left(-4a^{5}+2a^{4}+20a^{3}-6a^{2}-21a+3\right){x}-3a^{5}+3a^{4}+14a^{3}-10a^{2}-13a+1\)
sage: E = EllipticCurve([K([1,1,0,0,0,0]),K([5,-9,-14,13,4,-3]),K([-3,8,7,-9,-2,2]),K([3,-21,-6,20,2,-4]),K([1,-13,-10,14,3,-3])])
 
gp: E = ellinit([Polrev([1,1,0,0,0,0]),Polrev([5,-9,-14,13,4,-3]),Polrev([-3,8,7,-9,-2,2]),Polrev([3,-21,-6,20,2,-4]),Polrev([1,-13,-10,14,3,-3])], K);
 
magma: E := EllipticCurve([K![1,1,0,0,0,0],K![5,-9,-14,13,4,-3],K![-3,8,7,-9,-2,2],K![3,-21,-6,20,2,-4],K![1,-13,-10,14,3,-3]]);
 

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: \( -\frac{5980179215}{7} a^{5} + \frac{700090460}{7} a^{4} + \frac{21196892310}{7} a^{3} - 667291825 a^{2} - 1239034905 a + \frac{2088108851}{7} \)
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 $\left(4 a^{5} - 6 a^{4} - 19 a^{3} + 22 a^{2} + 19 a - 8 : -10 a^{5} + 14 a^{4} + 49 a^{3} - 50 a^{2} - 54 a + 15 : 1\right)$
Height \(0.0067901744378999781794857955067866795905\)
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: \( 0.0067901744378999781794857955067866795905 \)
Period: \( 41021.076722571213754526542822742893981 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.39946 \)
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-f 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.