Properties

Label 2.2.17.1-17.1-a3
Base field \(\Q(\sqrt{17}) \)
Conductor norm \( 17 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{17}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-4, -1, 1]))
 
gp: K = nfinit(Polrev([-4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(6a+9\right){x}+4a+6\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([9,6]),K([6,4])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([9,6]),Polrev([6,4])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![9,6],K![6,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a+1)\) = \((-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 17 \) = \(17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a+1)\) = \((-2a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -17 \) = \(-17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{5821794}{17} a + \frac{14936697}{17} \)
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 $\left(0 : a + 1 : 1\right)$
Height \(0.37940702221629152010400670040896841701\)
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: $\left(a + 2 : -4 a - 7 : 1\right)$
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: \( 0.37940702221629152010400670040896841701 \)
Period: \( 38.295292221564715715016301661017783066 \)
Tamagawa product: \( 1 \)
Torsion order: \(4\)
Leading coefficient: \( 0.44049025497952142301979342445866619099 \)
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\((j)_{-}\)
\((-2a+1)\) \(17\) \(1\) \(I_{1}\) 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, 4, 8 and 16.
Its isogeny class 17.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is a \(\Q\)-curve.

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