Properties

Label 4.4.2777.1-16.1-a4
Base field 4.4.2777.1
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.2777.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
 
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-3\right){x}^{2}+\left(248a^{3}+94a^{2}-178a-86\right){x}-61a^{3}+1602a^{2}-240a-924\)
sage: E = EllipticCurve([K([1,0,0,0]),K([-3,2,2,-1]),K([0,-4,0,1]),K([-86,-178,94,248]),K([-924,-240,1602,-61])])
 
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-3,2,2,-1]),Polrev([0,-4,0,1]),Polrev([-86,-178,94,248]),Polrev([-924,-240,1602,-61])], K);
 
magma: E := EllipticCurve([K![1,0,0,0],K![-3,2,2,-1],K![0,-4,0,1],K![-86,-178,94,248],K![-924,-240,1602,-61]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((-a)\cdot(a^3-a^2-4a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(2\cdot8\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((344a^3-2248a^2+1184a+4128)\) = \((-a)^{36}\cdot(a^3-a^2-4a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -35184372088832 \) = \(-2^{36}\cdot8^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{11596855953503300897687859}{68719476736} a^{3} + \frac{2020351217336280011001041}{68719476736} a^{2} + \frac{24027899520071110551507553}{34359738368} a + \frac{28086881208189762512317305}{68719476736} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{3}{4} a^{3} + \frac{13}{4} a^{2} - \frac{7}{2} a - 2 : -\frac{1}{8} a^{3} - \frac{13}{8} a^{2} + \frac{15}{4} a + 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 8.1323536889378345721522407358255358011 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 0.694449760441155 \)
Analytic order of Ш: \( 9 \) (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\((j)_{-}\)
\((-a)\) \(2\) \(2\) \(I_{36}\) Non-split multiplicative \(1\) \(1\) \(36\) \(36\)
\((a^3-a^2-4a+1)\) \(8\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 16.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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

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