Properties

Label 4.4.1600.1-1.1-a6
Base field \(\Q(\sqrt{2}, \sqrt{5})\)
Conductor norm \( 1 \)
CM no
Base change no
Q-curve yes
Torsion order \( 14 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{2}, \sqrt{5})\)

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

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

Weierstrass equation

\({y}^2+a{x}{y}+\left(\frac{1}{2}a^{2}+a-1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+2a-1\right){x}^{2}+\left(-a^{3}-a^{2}+a+1\right){x}-a^{3}-\frac{3}{2}a^{2}+a+1\)
sage: E = EllipticCurve([K([0,1,0,0]),K([-1,2,1/2,-1/2]),K([-1,1,1/2,0]),K([1,1,-1,-1]),K([1,1,-3/2,-1])])
 
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([-1,2,1/2,-1/2]),Polrev([-1,1,1/2,0]),Polrev([1,1,-1,-1]),Polrev([1,1,-3/2,-1])], K);
 
magma: E := EllipticCurve([K![0,1,0,0],K![-1,2,1/2,-1/2],K![-1,1,1/2,0],K![1,1,-1,-1],K![1,1,-3/2,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1 \) = 1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -313664 a^{3} + 717600 a^{2} + 241280 a - 548608 \)
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: \(\Z/14\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{1}{2} a^{3} - a^{2} - 3 a + 3 : \frac{3}{2} a^{3} - a^{2} - 7 a + 6 : 1\right)$
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: \( 2222.1339508952208809352129880113353638 \)
Tamagawa product: \( 1 \)
Torsion order: \(14\)
Leading coefficient: \( 0.283435452920309 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

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
\(7\) 7B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 7, 14, 21 and 42.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 42.

Base change

This elliptic curve is a \(\Q\)-curve.

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