Properties

Label 2.0.19.1-100.1-a1
Base field \(\Q(\sqrt{-19}) \)
Conductor norm \( 100 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

Generator \(a\), with minimal polynomial \( x^{2} - x + 5 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, -1, 1]))
 
gp: K = nfinit(Polrev([5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(29a+223\right){x}-697a+1006\)
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([0,0]),K([223,29]),K([1006,-697])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([0,0]),Polrev([223,29]),Polrev([1006,-697])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,1],K![0,0],K![223,29],K![1006,-697]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a-10)\) = \((2)\cdot(-a)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 100 \) = \(4\cdot5^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((171364032a+228852290)\) = \((2)\cdot(-a)^{24}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 238418579101562500 \) = \(4\cdot5^{24}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{546495468563548}{3814697265625} a - \frac{26594457793024591}{7629394531250} \)
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 $\left(5 a - 6 : -5 a + 28 : 1\right)$
Height \(1.9413033450858088134296114751241771120\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.9413033450858088134296114751241771120 \)
Period: \( 0.46590538537844622764682722743556955812 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 1.6599855969938814931587072824253524856 \)
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 \(1\) \(1\) \(1\) \(1\)
\((-a)\) \(5\) \(2\) \(I_{18}^{*}\) Additive \(1\) \(2\) \(24\) \(18\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 100.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

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

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