Properties

Label 2.2.76.1-18.1-c1
Base field \(\Q(\sqrt{19}) \)
Conductor norm \( 18 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6110a-26622\right){x}+269490a+1174670\)
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([0,1]),K([-26622,-6110]),K([1174670,269490])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([0,1]),Polrev([-26622,-6110]),Polrev([1174670,269490])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,1],K![0,1],K![-26622,-6110],K![1174670,269490]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-9a+39)\) = \((-3a+13)\cdot(-a-4)\cdot(-a+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 18 \) = \(2\cdot3\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((59520a+275712)\) = \((-3a+13)^{14}\cdot(-a-4)\cdot(-a+4)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 8707129344 \) = \(2^{14}\cdot3\cdot3^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \)
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 $\left(-a - 2 : 188 a + 821 : 1\right)$
Height \(0.13455558515816975440469531155628322567\)
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: $\left(23 a + 102 : -12 a - 51 : 1\right)$
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: \( 0.13455558515816975440469531155628322567 \)
Period: \( 10.288176656674997079774901209157094905 \)
Tamagawa product: \( 22 \)  =  \(2\cdot1\cdot11\)
Torsion order: \(2\)
Leading coefficient: \( 1.7467310128065948900132131700742232088 \)
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\((j)_{-}\)
\((-3a+13)\) \(2\) \(2\) \(I_{14}\) Non-split multiplicative \(1\) \(1\) \(14\) \(14\)
\((-a-4)\) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-a+4)\) \(3\) \(11\) \(I_{11}\) Split multiplicative \(-1\) \(1\) \(11\) \(11\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 18.1-c consists of curves linked by isogenies of degree 2.

Base change

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

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