Properties

Label 2.2.33.1-539.1-f3
Base field \(\Q(\sqrt{33}) \)
Conductor norm \( 539 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{33}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -1, 1]))
 
gp: K = nfinit(Polrev([-8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+{x}^{2}+441{x}-15815\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([441,0]),K([-15815,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([1,0]),Polrev([441,0]),Polrev([-15815,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![441,0],K![-15815,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-28a-63)\) = \((-4a-9)\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 539 \) = \(11\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-115539436859)\) = \((-4a-9)^{18}\cdot(7)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 13349361469694847785881 \) = \(11^{18}\cdot49^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{9463555063808}{115539436859} \)
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: \(1\)
Generator $\left(\frac{1967}{12} : -\frac{26551}{36} a + \frac{26515}{72} : 1\right)$
Height \(6.7060463979962524596942068978572326937\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 6.7060463979962524596942068978572326937 \)
Period: \( 0.26656915460277509195542130193897114993 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 2.4894847236012463273130410526192715515 \)
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\((j)_{-}\)
\((-4a-9)\) \(11\) \(2\) \(I_{18}\) Non-split multiplicative \(1\) \(1\) \(18\) \(18\)
\((7)\) \(49\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 539.1-f consists of curves linked by isogenies of degrees dividing 9.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 77.b3
\(\Q\) 7623.i3