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Properties

Label 4.4.13676.1-20.1-a1
Base field 4.4.13676.1
Conductor norm \( 20 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 4.4.13676.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + 7 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, -6, -1, 1]))
 
gp: K = nfinit(Polrev([1, 7, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, -6, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}-a^{2}+6a+1\right){x}^{2}+\left(-a^{3}+a^{2}+6a+1\right){x}+95a^{3}-107a^{2}-553a+737\)
sage: E = EllipticCurve([K([-1,-4,1,1]),K([1,6,-1,-1]),K([-3,0,1,0]),K([1,6,1,-1]),K([737,-553,-107,95])])
 
gp: E = ellinit([Polrev([-1,-4,1,1]),Polrev([1,6,-1,-1]),Polrev([-3,0,1,0]),Polrev([1,6,1,-1]),Polrev([737,-553,-107,95])], K);
 
magma: E := EllipticCurve([K![-1,-4,1,1],K![1,6,-1,-1],K![-3,0,1,0],K![1,6,1,-1],K![737,-553,-107,95]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2+a-5)\) = \((a^3+a^2-3a)\cdot(a^3-5a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 20 \) = \(4\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^2+a-5)\) = \((a^3+a^2-3a)\cdot(a^3-5a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -20 \) = \(-4\cdot5\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{85019}{10} a^{3} + \frac{95633}{10} a^{2} + \frac{493861}{10} a + \frac{61911}{10} \)
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 $\left(-a^{3} + a^{2} + 5 a - 7 : -2 a^{3} + 3 a^{2} + 13 a - 18 : 1\right)$
Height \(0.41381603161632405720617413484754565858\)
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: $\left(-a^{3} + \frac{5}{4} a^{2} + 5 a - \frac{35}{4} : \frac{7}{8} a^{3} + \frac{7}{8} a^{2} - \frac{25}{8} a - \frac{9}{8} : 1\right)$
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: \( 0.41381603161632405720617413484754565858 \)
Period: \( 1174.4130427789431496699722547202109723 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 4.15574251508528 \)
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\((j)_{-}\)
\((a^3+a^2-3a)\) \(4\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((a^3-5a+1)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 20.1-a 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.