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Properties

Label 6.6.1241125.1-45.1-b2
Base field 6.6.1241125.1
Conductor norm \( 45 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+\left(-2a^{5}+a^{4}+14a^{3}-a^{2}-23a-10\right){x}{y}+\left(-4a^{5}+2a^{4}+27a^{3}-4a^{2}-42a-12\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-3a^{2}-8a-3\right){x}^{2}+\left(132a^{5}-252a^{4}-589a^{3}+1066a^{2}+151a-199\right){x}-9857a^{5}+12020a^{4}+53933a^{3}-45477a^{2}-50903a-9271\)
sage: E = EllipticCurve([K([-10,-23,-1,14,1,-2]),K([-3,-8,-3,6,1,-1]),K([-12,-42,-4,27,2,-4]),K([-199,151,1066,-589,-252,132]),K([-9271,-50903,-45477,53933,12020,-9857])])
 
gp: E = ellinit([Polrev([-10,-23,-1,14,1,-2]),Polrev([-3,-8,-3,6,1,-1]),Polrev([-12,-42,-4,27,2,-4]),Polrev([-199,151,1066,-589,-252,132]),Polrev([-9271,-50903,-45477,53933,12020,-9857])], K);
 
magma: E := EllipticCurve([K![-10,-23,-1,14,1,-2],K![-3,-8,-3,6,1,-1],K![-12,-42,-4,27,2,-4],K![-199,151,1066,-589,-252,132],K![-9271,-50903,-45477,53933,12020,-9857]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-a^4-6a^3+3a^2+8a+4)\) = \((-2a^5+a^4+13a^3-2a^2-19a-5)\cdot(2a^5-a^4-14a^3+2a^2+23a+6)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 45 \) = \(5\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((41a^5-12a^4-264a^3-a^2+361a+164)\) = \((-2a^5+a^4+13a^3-2a^2-19a-5)^{8}\cdot(2a^5-a^4-14a^3+2a^2+23a+6)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2562890625 \) = \(5^{8}\cdot9^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{974480916610807874048}{50625} a^{5} + \frac{2192578635875788926082}{50625} a^{4} - \frac{377615335888840389773}{10125} a^{3} - \frac{2065709568086390135938}{16875} a^{2} - \frac{358246336659862446862}{5625} a - \frac{86620748507289493126}{10125} \)
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: $\left(-2 a^{5} - \frac{17}{4} a^{4} + \frac{83}{4} a^{3} + \frac{95}{4} a^{2} - \frac{197}{4} a - \frac{99}{4} : -\frac{39}{4} a^{5} - \frac{15}{8} a^{4} + \frac{297}{4} a^{3} + \frac{209}{8} a^{2} - \frac{1073}{8} a - \frac{495}{8} : 1\right)$
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: \( 60.149260071395482892383301727833290992 \)
Tamagawa product: \( 32 \)  =  \(2^{3}\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 1.72772 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((-2a^5+a^4+13a^3-2a^2-19a-5)\) \(5\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((2a^5-a^4-14a^3+2a^2+23a+6)\) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 and 4.
Its isogeny class 45.1-b consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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