Properties

Label 6.6.1241125.1-9.1-a1
Base field 6.6.1241125.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.1241125.1

Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
 
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(-a^{5}+7a^{3}+2a^{2}-11a-5\right){x}{y}+\left(-a^{5}+a^{4}+6a^{3}-3a^{2}-8a-3\right){y}={x}^{3}+\left(4a^{5}-2a^{4}-27a^{3}+3a^{2}+41a+15\right){x}^{2}+\left(18a^{5}+6a^{4}-105a^{3}-50a^{2}+97a+43\right){x}+47a^{5}+62a^{4}-218a^{3}-338a^{2}-56a+22\)
sage: E = EllipticCurve([K([-5,-11,2,7,0,-1]),K([15,41,3,-27,-2,4]),K([-3,-8,-3,6,1,-1]),K([43,97,-50,-105,6,18]),K([22,-56,-338,-218,62,47])])
 
gp: E = ellinit([Polrev([-5,-11,2,7,0,-1]),Polrev([15,41,3,-27,-2,4]),Polrev([-3,-8,-3,6,1,-1]),Polrev([43,97,-50,-105,6,18]),Polrev([22,-56,-338,-218,62,47])], K);
 
magma: E := EllipticCurve([K![-5,-11,2,7,0,-1],K![15,41,3,-27,-2,4],K![-3,-8,-3,6,1,-1],K![43,97,-50,-105,6,18],K![22,-56,-338,-218,62,47]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^5-a^4-14a^3+2a^2+23a+6)\) = \((2a^5-a^4-14a^3+2a^2+23a+6)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3a^5+a^4-19a^3-8a^2+23a+9)\) = \((2a^5-a^4-14a^3+2a^2+23a+6)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -729 \) = \(-9^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{79572442844}{27} a^{5} + \frac{179028656408}{27} a^{4} - \frac{154197900412}{27} a^{3} - \frac{506013769217}{27} a^{2} - \frac{263202231980}{27} a - \frac{35350337779}{27} \)
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: \( 1226.5360945033932874353896896020530865 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.10096 \)
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\((j)_{-}\)
\((2a^5-a^4-14a^3+2a^2+23a+6)\) \(9\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 9.1-a consists of curves linked by isogenies of degree 5.

Base change

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

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