Properties

Label 4.4.1600.1-31.1-a8
Base field \(\Q(\sqrt{2}, \sqrt{5})\)
Conductor norm \( 31 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+\left(\frac{1}{2}a^{3}-2a+1\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{3}+a\right){x}^{2}+\left(-38a^{3}+\frac{49}{2}a^{2}+185a-164\right){x}+125a^{3}-\frac{275}{2}a^{2}-686a+630\)
sage: E = EllipticCurve([K([1,-2,0,1/2]),K([0,1,0,-1/2]),K([1,0,0,0]),K([-164,185,49/2,-38]),K([630,-686,-275/2,125])])
 
gp: E = ellinit([Polrev([1,-2,0,1/2]),Polrev([0,1,0,-1/2]),Polrev([1,0,0,0]),Polrev([-164,185,49/2,-38]),Polrev([630,-686,-275/2,125])], K);
 
magma: E := EllipticCurve([K![1,-2,0,1/2],K![0,1,0,-1/2],K![1,0,0,0],K![-164,185,49/2,-38],K![630,-686,-275/2,125]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/2a^3+1/2a^2-3a)\) = \((1/2a^3+1/2a^2-3a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 31 \) = \(31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1/2a^3+1/2a^2-3a)\) = \((1/2a^3+1/2a^2-3a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 31 \) = \(31\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{115679900627333053956710625}{62} a^{3} - \frac{101107940561583547666668071}{62} a^{2} + \frac{302853911657568218576001528}{31} a + \frac{264704024922729858633502426}{31} \)
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: $\left(\frac{5}{4} a^{3} - 2 a^{2} - 8 a + \frac{37}{4} : -\frac{31}{16} a^{3} + \frac{5}{2} a^{2} + \frac{45}{4} a - \frac{85}{8} : 1\right)$
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: \( 170.63134034140379105605341726925923117 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 1.06644587713377 \)
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\((j)_{-}\)
\((1/2a^3+1/2a^2-3a)\) \(31\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 31.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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

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