Properties

Label 2.0.67.1-289.2-b1
Base field \(\Q(\sqrt{-67}) \)
Conductor \((17)\)
Conductor norm \( 289 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank not available

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Base field \(\Q(\sqrt{-67}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 17 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([17, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([17, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-a+17\right){x}\)
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([17,-1]),K([0,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([17,-1])),Pol(Vecrev([0,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![17,-1],K![0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((17)\) = \((-a)\cdot(a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 289 \) = \(17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-17a)\) = \((-a)^{2}\cdot(a-1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4913 \) = \(17^{2}\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{148207}{289} a - \frac{1202001}{289} \)
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: \(0 \le r \le 1\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(0 : -a : 1\right)$
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: \(0 \le r \le 1\)
Regulator: not available
Period: \( 6.34473030369887 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(4\)
Leading coefficient: \( 5.44074508706850 \)
Analytic order of Ш: not available

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\((j)_{-}\)
\((-a)\) \(17\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a-1)\) \(17\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 and 4.
Its isogeny class 289.2-b consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.