Properties

Label 6.6.485125.1-49.1-c1
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

\({y}^2+\left(-a^{5}+2a^{4}+5a^{3}-7a^{2}-6a+3\right){x}{y}+\left(a^{5}-5a^{3}+4a+1\right){y}={x}^{3}+\left(3a^{5}-4a^{4}-13a^{3}+13a^{2}+11a-5\right){x}^{2}+\left(4a^{5}-7a^{4}-19a^{3}+24a^{2}+23a-7\right){x}+3a^{5}-3a^{4}-15a^{3}+11a^{2}+14a-5\)
sage: E = EllipticCurve([K([3,-6,-7,5,2,-1]),K([-5,11,13,-13,-4,3]),K([1,4,0,-5,0,1]),K([-7,23,24,-19,-7,4]),K([-5,14,11,-15,-3,3])])
 
gp: E = ellinit([Polrev([3,-6,-7,5,2,-1]),Polrev([-5,11,13,-13,-4,3]),Polrev([1,4,0,-5,0,1]),Polrev([-7,23,24,-19,-7,4]),Polrev([-5,14,11,-15,-3,3])], K);
 
magma: E := EllipticCurve([K![3,-6,-7,5,2,-1],K![-5,11,13,-13,-4,3],K![1,4,0,-5,0,1],K![-7,23,24,-19,-7,4],K![-5,14,11,-15,-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: \((11a^5-15a^4-58a^3+65a^2+83a-47)\) = \((-2a^5+3a^4+10a^3-12a^2-11a+6)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 282475249 \) = \(49^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{795540805}{16807} a^{5} - \frac{1001826365}{16807} a^{4} - \frac{4000609051}{16807} a^{3} + \frac{3577994149}{16807} a^{2} + \frac{4448111321}{16807} a - \frac{1220726677}{16807} \)
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: $\left(-a^{5} + a^{4} + 5 a^{3} - 4 a^{2} - 5 a + 2 : -a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a : 1\right)$
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: \( 4621.4698728416095670049477649744600947 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 1.65880 \)
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\((j)_{-}\)
\((-2a^5+3a^4+10a^3-12a^2-11a+6)\) \(49\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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-c 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.