Properties

Label 3.3.733.1-5.1-a1
Base field 3.3.733.1
Conductor norm \( 5 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 3.3.733.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
 
gp: K = nfinit(Polrev([8, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(188a^{2}+287a-596\right){x}-218a^{2}-330a+693\)
sage: E = EllipticCurve([K([-4,0,1]),K([4,0,-1]),K([-5,1,1]),K([-596,287,188]),K([693,-330,-218])])
 
gp: E = ellinit([Polrev([-4,0,1]),Polrev([4,0,-1]),Polrev([-5,1,1]),Polrev([-596,287,188]),Polrev([693,-330,-218])], K);
 
magma: E := EllipticCurve([K![-4,0,1],K![4,0,-1],K![-5,1,1],K![-596,287,188],K![693,-330,-218]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a+3)\) = \((-a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5 \) = \(5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-24a^2-63a-35)\) = \((-a+3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1953125 \) = \(-5^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{18942486647612346}{1953125} a^{2} + \frac{70018873198097803}{1953125} a - \frac{56200973797329704}{1953125} \)
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 $\left(\frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4} : \frac{5}{8} a^{2} + \frac{19}{8} a - \frac{35}{8} : 1\right)$
Height \(2.3152498193711865143547353819607721709\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.3152498193711865143547353819607721709 \)
Period: \( 4.8795434198999187887172898463796631971 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.2518327616114287833579589682825554910 \)
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\((j)_{-}\)
\((-a+3)\) \(5\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 5.1-a consists of curves linked by isogenies of degree 3.

Base change

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

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