Properties

Base field 4.4.10512.1
Label 4.4.10512.1-4.1-a3
Conductor \((2,a^{3} - a^{2} - 5 a - 2)\)
Conductor norm \( 4 \)
CM no
base-change no
Q-curve no
Torsion order \( 3 \)
Rank not available

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Base field 4.4.10512.1

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

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 7*x^2 - 6*x + 1)
 
gp: K = nfinit(a^4 - 7*a^2 - 6*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);
 

Weierstrass equation

\( y^2 + \left(a^{2} - a - 4\right) x y + \left(a^{3} - 5 a - 5\right) y = x^{3} + a x^{2} + \left(29 a^{3} - 28 a^{2} - 147 a - 51\right) x + 12 a^{3} + 63 a^{2} - 206 a - 287 \)
sage: E = EllipticCurve(K, [a^2 - a - 4, a, a^3 - 5*a - 5, 29*a^3 - 28*a^2 - 147*a - 51, 12*a^3 + 63*a^2 - 206*a - 287])
 
gp: E = ellinit([a^2 - a - 4, a, a^3 - 5*a - 5, 29*a^3 - 28*a^2 - 147*a - 51, 12*a^3 + 63*a^2 - 206*a - 287],K)
 
magma: E := ChangeRing(EllipticCurve([a^2 - a - 4, a, a^3 - 5*a - 5, 29*a^3 - 28*a^2 - 147*a - 51, 12*a^3 + 63*a^2 - 206*a - 287]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((2,a^{3} - a^{2} - 5 a - 2)\) = \( \left(a^{3} - a^{2} - 5 a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 4 \) = \( 4 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((256,128 a^{3} - 128 a^{2} - 640 a,256 a,128 a^{2} - 128 a - 384)\) = \( \left(a^{3} - a^{2} - 5 a\right)^{15} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 1073741824 \) = \( 4^{15} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{266320696004661}{256} a^{3} + \frac{266320696004661}{256} a^{2} + \frac{1331603480023305}{256} a + \frac{90950209108247}{32} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(-2 a^{3} + a^{2} + 12 a + 11 : 5 a^{3} - 4 a^{2} - 28 a - 8 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

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\((j)_{-}\)
\( \left(a^{3} - a^{2} - 5 a\right) \) \(4\) \(15\) \(I_{15}\) Split multiplicative \(-1\) \(1\) \(15\) \(15\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Cs.1.1
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 4.1-a consists of curves linked by isogenies of degrees dividing 45.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.