Properties

Label 2.2.12.1-33.2-d2
Base field \(\Q(\sqrt{3}) \)
Conductor norm \( 33 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{3}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Polrev([-3, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(460939a-798370\right){x}+224206774a-388337524\)
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([1,0]),K([-798370,460939]),K([-388337524,224206774])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([1,0]),Polrev([-798370,460939]),Polrev([-388337524,224206774])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,0],K![1,0],K![-798370,460939],K![-388337524,224206774]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a-6)\) = \((a)\cdot(-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 33 \) = \(3\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((705a-204)\) = \((a)^{2}\cdot(-2a+1)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1449459 \) = \(-3^{2}\cdot11^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \)
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: $\left(\frac{421}{2} a - 365 : \frac{309}{4} a - \frac{535}{4} : 1\right)$
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: \( 0.72631039230780105735863478135527898530 \)
Tamagawa product: \( 10 \)  =  \(2\cdot5\)
Torsion order: \(2\)
Leading coefficient: \( 0.52416937564266454334599846207410082314 \)
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\((j)_{-}\)
\((a)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-2a+1)\) \(11\) \(5\) \(I_{5}\) 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
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 33.2-d consists of curves linked by isogenies of degrees dividing 10.

Base change

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

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