Properties

Label 4.4.1957.1-19.1-b1
Base field 4.4.1957.1
Conductor norm \( 19 \)
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 4.4.1957.1

Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -4, 0, 1]))
 
gp: K = nfinit(Polrev([1, -1, -4, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -4, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-1\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(38a^{3}-44a^{2}-62a-20\right){x}+401a^{3}-614a^{2}-483a+89\)
sage: E = EllipticCurve([K([-1,0,1,0]),K([-1,7,1,-2]),K([0,3,1,-1]),K([-20,-62,-44,38]),K([89,-483,-614,401])])
 
gp: E = ellinit([Polrev([-1,0,1,0]),Polrev([-1,7,1,-2]),Polrev([0,3,1,-1]),Polrev([-20,-62,-44,38]),Polrev([89,-483,-614,401])], K);
 
magma: E := EllipticCurve([K![-1,0,1,0],K![-1,7,1,-2],K![0,3,1,-1],K![-20,-62,-44,38],K![89,-483,-614,401]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-4a)\) = \((a^3-a^2-4a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 19 \) = \(19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((14a^3+20a^2-37a-65)\) = \((a^3-a^2-4a)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2476099 \) = \(19^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{22217245546666612500}{6859} a^{3} + \frac{8805551373160716923}{6859} a^{2} - \frac{85379003522040468141}{6859} a - \frac{56056237584037687000}{6859} \)
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: \( 1.1784379355932611540720436353635042110 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.665964854587993 \)
Analytic order of Ш: \( 25 \) (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\((j)_{-}\)
\((a^3-a^2-4a)\) \(19\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 19.1-b consists of curves linked by isogenies of degrees dividing 15.

Base change

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

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