Properties

Label 3.3.940.1-16.4-d3
Base field 3.3.940.1
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 3.3.940.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-4, -7, 0, 1]))
 
gp: K = nfinit(Polrev([-4, -7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+2a+5\right){x}^{2}+\left(776001989a^{2}-467700226a-5150128688\right){x}+37939505091840a^{2}-22866326787280a-251794887741124\)
sage: E = EllipticCurve([K([0,1,0]),K([5,2,-1]),K([-4,0,1]),K([-5150128688,-467700226,776001989]),K([-251794887741124,-22866326787280,37939505091840])])
 
gp: E = ellinit([Polrev([0,1,0]),Polrev([5,2,-1]),Polrev([-4,0,1]),Polrev([-5150128688,-467700226,776001989]),Polrev([-251794887741124,-22866326787280,37939505091840])], K);
 
magma: E := EllipticCurve([K![0,1,0],K![5,2,-1],K![-4,0,1],K![-5150128688,-467700226,776001989],K![-251794887741124,-22866326787280,37939505091840]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^2+a+4)\) = \((-a-2)^{3}\cdot(a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(2^{3}\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-64a+256)\) = \((-a-2)^{11}\cdot(a+1)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 8388608 \) = \(2^{11}\cdot2^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{11863}{64} a^{2} + \frac{9029}{8} a + \frac{245}{4} \)
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: $\left(-14381 a^{2} + 8667 a + 95443 : -4334 a^{2} + 2612 a + 28764 : 1\right)$
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: \( 55.434232296960275111735916984436376907 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 0.90403264838682468119908722817272121773 \)
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\((j)_{-}\)
\((-a-2)\) \(2\) \(1\) \(II^{*}\) Additive \(-1\) \(3\) \(11\) \(0\)
\((a+1)\) \(2\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

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 and 4.
Its isogeny class 16.4-d consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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