Properties

Label 2.0.163.1-400.1-a1
Base field \(\Q(\sqrt{-163}) \)
Conductor \((20)\)
Conductor norm \( 400 \)
CM no
Base change yes: 20a3
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2={x}^3+{x}^2-36{x}-140\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-36,0]),K([-140,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-36,0])),Pol(Vecrev([-140,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-36,0],K![-140,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((20)\) = \((2)^{2}\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 400 \) = \(4^{2}\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((4000000)\) = \((2)^{8}\cdot(5)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 16000000000000 \) = \(4^{8}\cdot25^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{20720464}{15625} \)
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: $\left(7 : 0 : 1\right)$
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: \( 1.07051594281336 \)
Tamagawa product: \( 18 \)  =  \(3\cdot( 2 \cdot 3 )\)
Torsion order: \(2\)
Leading coefficient: \( 0.754643519178544 \)
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\((j)_{-}\)
\((2)\) \(4\) \(3\) \(IV^{*}\) Additive \(1\) \(2\) \(8\) \(0\)
\((5)\) \(25\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 400.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base change of 20a3, defined over \(\Q\), so it is also a \(\Q\)-curve.