Properties

Label 2.2.217.1-28.1-d1
Base field \(\Q(\sqrt{217}) \)
Conductor norm \( 28 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{217}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 54 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-54, -1, 1]))
 
gp: K = nfinit(Polrev([-54, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-54, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-301955a-2073057\right){x}+1763011434a+12103884350\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,1]),K([-2073057,-301955]),K([12103884350,1763011434])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,1]),Polrev([-2073057,-301955]),Polrev([12103884350,1763011434])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,1],K![-2073057,-301955],K![12103884350,1763011434]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-996a+7834)\) = \((-a+8)\cdot(-a-7)\cdot(-498a+3917)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28 \) = \(2\cdot2\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1344a-11200)\) = \((-a+8)^{6}\cdot(-a-7)^{12}\cdot(-498a+3917)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 12845056 \) = \(2^{6}\cdot2^{12}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{3360549}{28672} a + \frac{12727843}{14336} \)
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 $\left(-179 a - 1229 : -9358 a - 64244 : 1\right)$
Height \(0.14906175771795617237507129230107754112\)
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: $\left(-\frac{939}{4} a - \frac{6447}{4} : \frac{935}{8} a + \frac{6443}{8} : 1\right)$
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: \( 0.14906175771795617237507129230107754112 \)
Period: \( 9.6524497633838462667163221756553188592 \)
Tamagawa product: \( 48 \)  =  \(2\cdot( 2^{2} \cdot 3 )\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.3441487289522040673643404290047398609 \)
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\((j)_{-}\)
\((-a+8)\) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-a-7)\) \(2\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((-498a+3917)\) \(7\) \(2\) \(I_{2}\) Non-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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 28.1-d consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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