Properties

Label 5.5.14641.1-43.1-a2
Base field \(\Q(\zeta_{11})^+\)
Conductor \((-a^3-a^2+2a+3)\)
Conductor norm \( 43 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{11})^+\)

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -4, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-1, 3, 3, -4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-4a^{2}+3\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{4}+a^{3}+3a^{2}-2a-1\right){x}^{2}+\left(294a^{4}-510a^{3}-733a^{2}+1419a-379\right){x}+4714a^{4}-8577a^{3}-11707a^{2}+23701a-5755\)
sage: E = EllipticCurve([K([3,0,-4,0,1]),K([-1,-2,3,1,-1]),K([0,-2,0,1,0]),K([-379,1419,-733,-510,294]),K([-5755,23701,-11707,-8577,4714])])
 
gp: E = ellinit([Pol(Vecrev([3,0,-4,0,1])),Pol(Vecrev([-1,-2,3,1,-1])),Pol(Vecrev([0,-2,0,1,0])),Pol(Vecrev([-379,1419,-733,-510,294])),Pol(Vecrev([-5755,23701,-11707,-8577,4714]))], K);
 
magma: E := EllipticCurve([K![3,0,-4,0,1],K![-1,-2,3,1,-1],K![0,-2,0,1,0],K![-379,1419,-733,-510,294],K![-5755,23701,-11707,-8577,4714]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3-a^2+2a+3)\) = \((-a^3-a^2+2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 43 \) = \(43\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-58a^4+83a^3+183a^2-349a-31)\) = \((-a^3-a^2+2a+3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -271818611107 \) = \(-43^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{318410900614494023095122767009}{271818611107} a^{4} + \frac{98618610535417258075069052493}{271818611107} a^{3} - \frac{1144480542817566448101585501712}{271818611107} a^{2} - \frac{543717833889392347185233669478}{271818611107} a + \frac{243113372524798476732529529704}{271818611107} \)
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: \( 0.339304693114637 \)
Tamagawa product: \( 7 \)
Torsion order: \(1\)
Leading coefficient: \( 0.961830658994384 \)
Analytic order of Ш: \( 49 \) (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\((j)_{-}\)
\((-a^3-a^2+2a+3)\) \(43\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 43.1-a consists of curves linked by isogenies of degree 7.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.