Properties

Label 2.2.205.1-45.1-b3
Base field \(\Q(\sqrt{205}) \)
Conductor \((3a+21)\)
Conductor norm \( 45 \)
CM no
Base change yes: 25215.f8,75.b8
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

\({y}^2+\left(w+1\right){x}{y}={x}^3+\left(4519w+30113\right){x}-206840w-1377310\)
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([30113,4519]),K([-1377310,-206840])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([30113,4519])),Pol(Vecrev([-1377310,-206840]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![30113,4519],K![-1377310,-206840]]);
 

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: \((-3515625)\) = \((3,a)^{2}\cdot(3,a+2)^{2}\cdot(a+7)^{16}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 12359619140625 \) = \(3^{2}\cdot3^{2}\cdot5^{16}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{4733169839}{3515625} \)
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 $\left(\frac{41}{3} a + \frac{796}{9} : \frac{2104}{9} a + \frac{41834}{27} : 1\right)$
Height \(1.35303296720054\)
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: $\left(\frac{13}{4} a + 19 : -\frac{51}{4} a - \frac{739}{8} : 1\right)$
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: \( 64 \)  =  \(2\cdot2\cdot2^{4}\)
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\((j)_{-}\)
\((3,a)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((3,a+2)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a+7)\) \(5\) \(16\) \(I_{16}\) Split multiplicative \(-1\) \(1\) \(16\) \(16\)

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 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.f8, 75.b8, defined over \(\Q\), so it is also a \(\Q\)-curve.