Properties

Label 2.0.67.1-153.1-a1
Base field \(\Q(\sqrt{-67}) \)
Conductor \((-3a)\)
Conductor norm \( 153 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-3a-1\right){x}-5a+16\)
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,1]),K([-1,-3]),K([16,-5])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([-1,-3])),Pol(Vecrev([16,-5]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,1],K![-1,-3],K![16,-5]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a)\) = \((3)\cdot(-a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 153 \) = \(9\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8019a+66096)\) = \((3)^{5}\cdot(-a)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4931831529 \) = \(9^{5}\cdot17^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{232667875}{6765201} a + \frac{294541375}{20295603} \)
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 $\left(-a : -13 : 1\right)$
Height \(0.178252297614374\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.178252297614374 \)
Period: \( 2.17153248855038 \)
Tamagawa product: \( 20 \)  =  \(5\cdot2^{2}\)
Torsion order: \(1\)
Leading coefficient: \( 3.78315428758061 \)
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\((j)_{-}\)
\((3)\) \(9\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((-a)\) \(17\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 153.1-a consists of this curve only.

Base change

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