Properties

Label 2.2.205.1-45.1-b6
Base field \(\Q(\sqrt{205}) \)
Conductor \((3a+21)\)
Conductor norm \( 45 \)
CM no
Base change yes: 25215.f2,75.b2
Q-curve yes
Torsion order \( 4 \)
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

\({y}^2+w{x}{y}={x}^3+\left(-w+1\right){x}^2+\left(17411w-133328\right){x}+3454968w-26461272\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([-133328,17411]),K([-26461272,3454968])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-133328,17411])),Pol(Vecrev([-26461272,3454968]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![-133328,17411],K![-26461272,3454968]]);
 

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: \((164025)\) = \((3,a)^{8}\cdot(3,a+2)^{8}\cdot(a+7)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 26904200625 \) = \(3^{8}\cdot3^{8}\cdot5^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{272223782641}{164025} \)
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 $\left(-\frac{421}{9} a + \frac{3199}{9} : -\frac{21985}{27} a + \frac{168673}{27} : 1\right)$
Height \(1.35303296720054\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{83}{4} a - \frac{647}{4} : \frac{141}{2} a - \frac{4233}{8} : 1\right)$ $\left(20 a - 156 : 68 a - 510 : 1\right)$
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: \( 1.35303296720054 \)
Period: \( 1.96168888219134 \)
Tamagawa product: \( 256 \)  =  \(2^{3}\cdot2^{3}\cdot2^{2}\)
Torsion order: \(4\)
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\((j)_{-}\)
\((3,a)\) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((3,a+2)\) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((a+7)\) \(5\) \(4\) \(I_{4}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
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.f2, 75.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.