Properties

Label 2.2.205.1-45.1-b2
Base field \(\Q(\sqrt{205}) \)
Conductor \((3a+21)\)
Conductor norm \( 45 \)
CM no
Base change yes: 25215.f7,75.b7
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{205}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 51 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-51, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-51, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51, -1, 1]);
 

Weierstrass equation

\({y}^2+w{x}{y}={x}^3+\left(-w+1\right){x}^2+\left(-4w+52\right){x}-1032w+7923\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([52,-4]),K([7923,-1032])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([52,-4])),Pol(Vecrev([7923,-1032]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![52,-4],K![7923,-1032]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3a+21)\) = \((3,a)\cdot(3,a+2)\cdot(a+7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 45 \) = \(3\cdot3\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-15)\) = \((3,a)\cdot(3,a+2)\cdot(a+7)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 225 \) = \(3\cdot3\cdot5^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1}{15} \)
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 $\left(-\frac{94}{441} a - \frac{449}{441} : \frac{84467}{9261} a - \frac{552947}{9261} : 1\right)$
Height \(2.70606593440109\)
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: $\left(2 a - 18 : 8 a - 51 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.70606593440109 \)
Period: \( 31.3870221150614 \)
Tamagawa product: \( 2 \)  =  \(1\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 5.93214225492112 \)
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\((j)_{-}\)
\((3,a)\) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((3,a+2)\) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((a+7)\) \(5\) \(2\) \(I_{2}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 45.1-b consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of 25215.f7, 75.b7, defined over \(\Q\), so it is also a \(\Q\)-curve.