Properties

Label 4.4.4205.1-7.2-a3
Base field 4.4.4205.1
Conductor norm \( 7 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.4205.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, -1, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-5a\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(-a^{2}+3a+3\right){x}^{2}+\left(-17a^{3}+28a^{2}+64a-29\right){x}+39a^{3}-63a^{2}-164a+42\)
sage: E = EllipticCurve([K([0,-5,-1,1]),K([3,3,-1,0]),K([-1,-5,-1,1]),K([-29,64,28,-17]),K([42,-164,-63,39])])
 
gp: E = ellinit([Polrev([0,-5,-1,1]),Polrev([3,3,-1,0]),Polrev([-1,-5,-1,1]),Polrev([-29,64,28,-17]),Polrev([42,-164,-63,39])], K);
 
magma: E := EllipticCurve([K![0,-5,-1,1],K![3,3,-1,0],K![-1,-5,-1,1],K![-29,64,28,-17],K![42,-164,-63,39]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-2a-3)\) = \((a^2-2a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 7 \) = \(7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a^3-3a^2-7a+1)\) = \((a^2-2a-3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -49 \) = \(-7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{324563227463686700}{49} a^{3} + \frac{210301973473262610}{49} a^{2} + \frac{1696852149970179035}{49} a + \frac{921933595004884184}{49} \)
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: $\left(\frac{7}{4} a^{3} - \frac{5}{2} a^{2} - 8 a - \frac{1}{4} : -\frac{5}{4} a^{3} + \frac{5}{2} a^{2} + \frac{37}{8} a - \frac{7}{2} : 1\right)$
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: \( 153.00284755379781144913281774817510188 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.17974057890114 \)
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\((j)_{-}\)
\((a^2-2a-3)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 7.2-a consists of curves linked by isogenies of degrees dividing 10.

Base change

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

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