Properties

Label 2.2.76.1-200.1-b2
Base field \(\Q(\sqrt{19}) \)
Conductor norm \( 200 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{19}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, 0, 1]))
 
gp: K = nfinit(Polrev([-19, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(23211a-101122\right){x}+3377115a-14720430\)
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([0,0]),K([-101122,23211]),K([-14720430,3377115])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,0]),Polrev([-101122,23211]),Polrev([-14720430,3377115])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,0],K![-101122,23211],K![-14720430,3377115]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-30a+130)\) = \((-3a+13)^{3}\cdot(2a+9)\cdot(-2a+9)\)
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: \((100)\) = \((-3a+13)^{4}\cdot(2a+9)^{2}\cdot(-2a+9)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 10000 \) = \(2^{4}\cdot5^{2}\cdot5^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{148176}{25} \)
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 $\left(\frac{6351}{289} a - \frac{28892}{289} : \frac{1106978}{4913} a - \frac{4770151}{4913} : 1\right)$
Height \(2.3129186379494205363524573979494949232\)
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: $\left(\frac{77}{2} a - 172 : \frac{267}{4} a - \frac{1119}{4} : 1\right)$ $\left(19 a - 87 : 34 a - 137 : 1\right)$
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: \( 2.3129186379494205363524573979494949232 \)
Period: \( 17.627843130015737339377496613397455149 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 4.6768424148351217317286344278021625517 \)
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\((j)_{-}\)
\((-3a+13)\) \(2\) \(2\) \(III\) Additive \(-1\) \(3\) \(4\) \(0\)
\((2a+9)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-2a+9)\) \(5\) \(2\) \(I_{2}\) 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-b consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 40.a2
\(\Q\) 28880.s2