Properties

Label 2.0.11.1-2500.3-g1
Base field \(\Q(\sqrt{-11}) \)
Conductor norm \( 2500 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{-11}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
 
gp: K = nfinit(Polrev([3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a-48\right){x}+80a-226\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([-48,-2]),K([-226,80])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([-48,-2]),Polrev([-226,80])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![-48,-2],K![-226,80]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((50)\) = \((2)\cdot(-a-1)^{2}\cdot(a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2500 \) = \(4\cdot5^{2}\cdot5^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((11718750a-7812500)\) = \((2)\cdot(-a-1)^{11}\cdot(a-2)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 381469726562500 \) = \(4\cdot5^{11}\cdot5^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2632683}{6250} a + \frac{1560961}{6250} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.83782872227173162506165069723282725849 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 2.0209189166241559457352627528658977305 \)
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\((j)_{-}\)
\((2)\) \(4\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-a-1)\) \(5\) \(2\) \(I_{5}^{*}\) Additive \(1\) \(2\) \(11\) \(5\)
\((a-2)\) \(5\) \(2\) \(III^{*}\) 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
\(5\) 5B.1.4

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 2500.3-g consists of curves linked by isogenies of degree 5.

Base change

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

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