Properties

Label 4.4.2000.1-320.1-c2
Base field \(\Q(\zeta_{20})^+\)
Conductor norm \( 320 \)
CM no
Base change yes
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\zeta_{20})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, 0, -5, 0, 1]))
 
gp: K = nfinit(Polrev([5, 0, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+4\right){x}^{2}+\left(-133a^{3}-286a^{2}+562a+856\right){x}+2304a^{3}+1792a^{2}-7692a-7760\)
sage: E = EllipticCurve([K([-3,1,1,0]),K([4,-4,-1,1]),K([-3,1,1,0]),K([856,562,-286,-133]),K([-7760,-7692,1792,2304])])
 
gp: E = ellinit([Polrev([-3,1,1,0]),Polrev([4,-4,-1,1]),Polrev([-3,1,1,0]),Polrev([856,562,-286,-133]),Polrev([-7760,-7692,1792,2304])], K);
 
magma: E := EllipticCurve([K![-3,1,1,0],K![4,-4,-1,1],K![-3,1,1,0],K![856,562,-286,-133],K![-7760,-7692,1792,2304]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^2-2a+10)\) = \((a^3-a^2-3a+2)^{3}\cdot(a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 320 \) = \(4^{3}\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((160)\) = \((a^3-a^2-3a+2)^{10}\cdot(a)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 655360000 \) = \(4^{10}\cdot5^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{9285883494578}{5} a^{2} - \frac{12832775369604}{5} \)
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 $\left(-12 a^{3} - 2 a^{2} + 34 a + 22 : -42 a^{3} - 97 a^{2} + 187 a + 287 : 1\right)$
Height \(0.76325225226125977908508157769865810372\)
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: $\left(-\frac{3}{2} a^{3} - 5 a^{2} + \frac{15}{2} a + 11 : \frac{1}{4} a^{3} - a^{2} + \frac{3}{2} a + \frac{7}{4} : 1\right)$
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: \( 0.76325225226125977908508157769865810372 \)
Period: \( 5.1247543767085294550161754821412523171 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 2.79882748443601 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((a^3-a^2-3a+2)\) \(4\) \(2\) \(III^{*}\) Additive \(1\) \(3\) \(10\) \(0\)
\((a)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 320.1-c consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q(\sqrt{5}) \) 2.2.5.1-320.1-b6
\(\Q(\sqrt{5}) \) a curve with conductor norm 6400 (not in the database)