Properties

Label 2.0.163.1-604.1-a1
Base field \(\Q(\sqrt{-163}) \)
Conductor \((2a+20)\)
Conductor norm \( 604 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{-163}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 41 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([41, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^3+{x}^2+\left(a-2\right){x}+2a+16\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([0,1]),K([-2,1]),K([16,2])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-2,1])),Pol(Vecrev([16,2]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![0,1],K![-2,1],K![16,2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a+20)\) = \((2)\cdot(a+10)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 604 \) = \(4\cdot151\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2688a-7552)\) = \((2)^{7}\cdot(a+10)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 373571584 \) = \(4^{7}\cdot151^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2066959707}{2918528} a - \frac{868213747}{364816} \)
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: \(2\)
Generators $\left(2 : -a - 6 : 1\right)$ $\left(-\frac{2178692648}{2916756049} a + \frac{12181693989}{2916756049} : -\frac{348813732629960}{157525243938343} a + \frac{350035888574377}{157525243938343} : 1\right)$
Heights \(0.323292083817366\) \(11.7482499025556\)
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: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 3.66863840704340 \)
Period: \( 2.48255271670743 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 11.4137818661094 \)
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\((j)_{-}\)
\((2)\) \(4\) \(1\) \(I_{7}\) Non-split multiplicative \(1\) \(1\) \(7\) \(7\)
\((a+10)\) \(151\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 604.1-a consists of this curve only.

Base change

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