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Properties

Label 6.6.485125.1-49.1-a1
Base field 6.6.485125.1
Conductor norm \( 49 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+\left(-a^{5}+2a^{4}+4a^{3}-7a^{2}-3a+3\right){x}{y}+\left(a^{4}-a^{3}-3a^{2}+3a+2\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+3a^{2}+5a-1\right){x}^{2}+\left(4a^{5}-8a^{4}-16a^{3}+32a^{2}+10a-24\right){x}-21a^{5}+37a^{4}+92a^{3}-148a^{2}-74a+90\)
sage: E = EllipticCurve([K([3,-3,-7,4,2,-1]),K([-1,5,3,-5,-1,1]),K([2,3,-3,-1,1,0]),K([-24,10,32,-16,-8,4]),K([90,-74,-148,92,37,-21])])
 
gp: E = ellinit([Polrev([3,-3,-7,4,2,-1]),Polrev([-1,5,3,-5,-1,1]),Polrev([2,3,-3,-1,1,0]),Polrev([-24,10,32,-16,-8,4]),Polrev([90,-74,-148,92,37,-21])], K);
 
magma: E := EllipticCurve([K![3,-3,-7,4,2,-1],K![-1,5,3,-5,-1,1],K![2,3,-3,-1,1,0],K![-24,10,32,-16,-8,4],K![90,-74,-148,92,37,-21]]);
 

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: \( \frac{1287545907}{7} a^{5} - \frac{1693566461}{7} a^{4} - \frac{6310424038}{7} a^{3} + \frac{5984983744}{7} a^{2} + \frac{6664599078}{7} a - \frac{1884723034}{7} \)
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: \(0\)
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: $\left(2 a^{5} - 4 a^{4} - 9 a^{3} + 16 a^{2} + 9 a - 9 : -2 a^{5} + 3 a^{4} + 9 a^{3} - 12 a^{2} - 8 a + 7 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 4658.7283551974310651007958669273336718 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 1.67217 \)
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\((j)_{-}\)
\((-2a^5+3a^4+10a^3-12a^2-11a+6)\) \(49\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 49.1-a 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.