Properties

Label 2.2.17.1-324.1-f3
Base field \(\Q(\sqrt{17}) \)
Conductor \((18)\)
Conductor norm \( 324 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
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(Pol(Vecrev([-4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(67a-177\right){x}+776a-1991\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([-177,67]),K([-1991,776])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-177,67])),Pol(Vecrev([-1991,776]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![-177,67],K![-1991,776]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((18)\) = \((-a+2)\cdot(-a-1)\cdot(3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 324 \) = \(2\cdot2\cdot9^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1180980a+4094064)\) = \((-a+2)^{12}\cdot(-a-1)^{2}\cdot(3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -6347497291776 \) = \(-2^{12}\cdot2^{2}\cdot9^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5106375}{4096} a + \frac{4762125}{4096} \)
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 $\left(-9 a + 26 : -50 a + 123 : 1\right)$
Height \(1.42315639731243\)
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: $\left(-\frac{21}{4} a + 13 : \frac{17}{8} a - \frac{13}{2} : 1\right)$
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: \( 1.42315639731243 \)
Period: \( 1.87436795872976 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.58787331065193 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\((-a-1)\) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((3)\) \(9\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)

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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 324.1-f consists of curves linked by isogenies of degrees dividing 6.

Base change

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