Properties

Label 2.2.17.1-324.1-a3
Base field \(\Q(\sqrt{17}) \)
Conductor \((18)\)
Conductor norm \( 324 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-33a-39\right){x}+131a+113\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([-39,-33]),K([113,131])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-39,-33])),Pol(Vecrev([113,131]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-39,-33],K![113,131]]);
 

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: \((11967264a+15186528)\) = \((-a+2)^{5}\cdot(-a-1)^{20}\cdot(3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -160489808068608 \) = \(-2^{5}\cdot2^{20}\cdot9^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3440210411}{3145728} a + \frac{1389741023}{786432} \)
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: \(0\)
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: $\left(\frac{5}{4} a + \frac{23}{4} : -\frac{5}{8} a - \frac{23}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2.32501256143308 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 0.563898374804497 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((-a-1)\) \(2\) \(2\) \(I_{20}\) Non-split multiplicative \(1\) \(1\) \(20\) \(20\)
\((3)\) \(9\) \(2\) \(I_1^{*}\) Additive \(1\) \(2\) \(7\) \(1\)

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
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 5, 10 and 20.
Its isogeny class 324.1-a consists of curves linked by isogenies of degrees dividing 20.

Base change

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