Properties

Label 4.4.10512.1-4.1-a3
Base field 4.4.10512.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 1 \)

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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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -7, 0, 1]))
 
gp: K = nfinit(Polrev([1, -6, -7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{3}-5a-5\right){y}={x}^{3}+a{x}^{2}+\left(29a^{3}-28a^{2}-147a-51\right){x}+12a^{3}+63a^{2}-206a-287\)
sage: E = EllipticCurve([K([-4,-1,1,0]),K([0,1,0,0]),K([-5,-5,0,1]),K([-51,-147,-28,29]),K([-287,-206,63,12])])
 
gp: E = ellinit([Polrev([-4,-1,1,0]),Polrev([0,1,0,0]),Polrev([-5,-5,0,1]),Polrev([-51,-147,-28,29]),Polrev([-287,-206,63,12])], K);
 
magma: E := EllipticCurve([K![-4,-1,1,0],K![0,1,0,0],K![-5,-5,0,1],K![-51,-147,-28,29],K![-287,-206,63,12]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-5a)\) = \((a^3-a^2-5a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((128a^3-128a^2-640a)\) = \((a^3-a^2-5a)^{15}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1073741824 \) = \(4^{15}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{266320696004661}{256} a^{3} + \frac{266320696004661}{256} a^{2} + \frac{1331603480023305}{256} a + \frac{90950209108247}{32} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
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 $\left(\frac{90202}{64009} a^{3} - \frac{3771}{64009} a^{2} - \frac{662608}{64009} a - \frac{302029}{64009} : \frac{121612401}{16194277} a^{3} - \frac{141522592}{16194277} a^{2} - \frac{702015104}{16194277} a - \frac{57679796}{16194277} : 1\right)$
Height \(1.0758218114330386365153533349269547945\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion 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)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.0758218114330386365153533349269547945 \)
Period: \( 28.659471545381856813935013773342846714 \)
Tamagawa product: \( 15 \)
Torsion order: \(3\)
Leading coefficient: \( 2.00481631535149 \)
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\((j)_{-}\)
\((a^3-a^2-5a)\) \(4\) \(15\) \(I_{15}\) 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 elliptic curve is not a \(\Q\)-curve.

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