Properties

Label 2.2.17.1-612.1-l1
Base field \(\Q(\sqrt{17}) \)
Conductor norm \( 612 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 6 \)
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

\({y}^2+{x}{y}+{y}={x}^{3}-216{x}+2062\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-216,0]),K([2062,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-216,0]),Polrev([2062,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-216,0],K![2062,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-12a+6)\) = \((-a+2)\cdot(-a-1)\cdot(3)\cdot(-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 612 \) = \(2\cdot2\cdot9\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1228691592)\) = \((-a+2)^{3}\cdot(-a-1)^{3}\cdot(3)^{12}\cdot(-2a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1509683028251494464 \) = \(2^{3}\cdot2^{3}\cdot9^{12}\cdot17^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1107111813625}{1228691592} \)
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 $\left(-\frac{128}{17} : -\frac{8019}{289} a + \frac{4953}{289} : 1\right)$
Height \(2.6706039824343995796898097660263012620\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(20 : 66 : 1\right)$
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: \( 2.6706039824343995796898097660263012620 \)
Period: \( 1.9405395242985687019079262778741845769 \)
Tamagawa product: \( 24 \)  =  \(1\cdot1\cdot( 2^{2} \cdot 3 )\cdot2\)
Torsion order: \(6\)
Leading coefficient: \( 1.6758928995864254431083541229513363505 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((-a-1)\) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((3)\) \(9\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((-2a+1)\) \(17\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 612.1-l consists of curves linked by isogenies of degrees dividing 12.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 102.b3
\(\Q\) 1734.b3