Properties

Label 2.2.44.1-200.1-h2
Base field \(\Q(\sqrt{11}) \)
Conductor norm \( 200 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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

Generator \(a\), with minimal polynomial \( x^{2} - 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-11, 0, 1]))
 
gp: K = nfinit(Polrev([-11, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-9a-26\right){x}+13a+44\)
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([0,0]),K([-26,-9]),K([44,13])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-26,-9]),Polrev([44,13])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![0,0],K![-26,-9],K![44,13]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((10a+30)\) = \((a+3)^{3}\cdot(a-4)\cdot(-a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 200 \) = \(2^{3}\cdot5\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((100a+600)\) = \((a+3)^{4}\cdot(a-4)^{4}\cdot(-a-4)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 250000 \) = \(2^{4}\cdot5^{4}\cdot5^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{70243008}{625} a + \frac{234600848}{625} \)
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 $\left(-\frac{11}{49} a - \frac{118}{49} : \frac{1244}{343} a + \frac{3469}{343} : 1\right)$
Height \(1.0471140478444986109961457307996788890\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-a - 5 : 3 a + 8 : 1\right)$ $\left(\frac{1}{2} a : -\frac{1}{4} a - \frac{11}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.0471140478444986109961457307996788890 \)
Period: \( 11.569470871288166147543226549353317067 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 3.6526759103659143099258495796207994315 \)
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\((j)_{-}\)
\((a+3)\) \(2\) \(2\) \(III\) Additive \(-1\) \(3\) \(4\) \(0\)
\((a-4)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-a-4)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 200.1-h consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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