Properties

Label 2.2.497.1-4.1-d1
Base field \(\Q(\sqrt{497}) \)
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{497}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 124 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-124, -1, 1]))
 
gp: K = nfinit(Polrev([-124, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-124, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(13671a-159228\right){x}-611457a+7121452\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([0,1]),K([-159228,13671]),K([7121452,-611457])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([0,1]),Polrev([-159228,13671]),Polrev([7121452,-611457])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![0,1],K![-159228,13671],K![7121452,-611457]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((17a-198)\cdot(17a+181)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(2\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-13432a+156440)\) = \((17a-198)^{3}\cdot(17a+181)^{15}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 262144 \) = \(2^{3}\cdot2^{15}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2663311}{32768} a + \frac{21913717}{8192} \)
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 $\left(4 a - 47 : 282 a - 3290 : 1\right)$
Height \(0.17383815170964199971034107340706973138\)
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: \( 0.17383815170964199971034107340706973138 \)
Period: \( 22.064129172836300213727081559521388475 \)
Tamagawa product: \( 15 \)  =  \(1\cdot( 3 \cdot 5 )\)
Torsion order: \(1\)
Leading coefficient: \( 5.1614883038477973820792859216003150172 \)
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\((j)_{-}\)
\((17a-198)\) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((17a+181)\) \(2\) \(15\) \(I_{15}\) Split multiplicative \(-1\) \(1\) \(15\) \(15\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 4.1-d consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

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