Properties

Label 2.2.205.1-28.1-a2
Base field \(\Q(\sqrt{205}) \)
Conductor norm \( 28 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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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(Polrev([-51, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(2754w-21093\right){x}+209802w-1606855\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-21093,2754]),K([-1606855,209802])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-21093,2754]),Polrev([-1606855,209802])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-21093,2754],K![-1606855,209802]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((14,2a+2)\) = \((2)\cdot(7,a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28 \) = \(4\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1284a+12652)\) = \((2)^{2}\cdot(7,a+1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 92236816 \) = \(4^{2}\cdot7^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{780348831047775}{23059204} a + \frac{5976664389642267}{23059204} \)
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 $\left(-\frac{728176}{14161} a + \frac{5583563}{14161} : \frac{2299198816}{1685159} a - \frac{17610618090}{1685159} : 1\right)$
Height \(7.1845599415850338546168916636052085422\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(8 a - \frac{245}{4} : -4 a + \frac{241}{8} : 1\right)$ $\left(8 a - 61 : -4 a + 30 : 1\right)$
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: \( 7.1845599415850338546168916636052085422 \)
Period: \( 3.9432131433469299769855676139040918823 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 3.9573411436864448972454826518406372622 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((2)\) \(4\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((7,a+1)\) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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.
Its isogeny class 28.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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