Properties

Label 3.3.316.1-64.7-d1
Base field 3.3.316.1
Conductor \((2a^2-3a-1)\)
Conductor norm \( 64 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field 3.3.316.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, -4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(7908273a^{2}-4185996a-33603392\right){x}-17708327376a^{2}+9373311329a+75245171793\)
sage: E = EllipticCurve([K([-3,0,1]),K([-4,-1,1]),K([-3,0,1]),K([-33603392,-4185996,7908273]),K([75245171793,9373311329,-17708327376])])
 
gp: E = ellinit([Pol(Vecrev([-3,0,1])),Pol(Vecrev([-4,-1,1])),Pol(Vecrev([-3,0,1])),Pol(Vecrev([-33603392,-4185996,7908273])),Pol(Vecrev([75245171793,9373311329,-17708327376]))], K);
 
magma: E := EllipticCurve([K![-3,0,1],K![-4,-1,1],K![-3,0,1],K![-33603392,-4185996,7908273],K![75245171793,9373311329,-17708327376]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^2-3a-1)\) = \((-a+1)^{6}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 64 \) = \(2^{6}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-15a^2+46a+93)\) = \((-a+1)^{19}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -524288 \) = \(2^{19}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{10154621765056003}{2} a^{2} + 2687505089603077 a + 21574207961612782 \)
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 $\left(-\frac{8361965}{10658} a^{2} + \frac{2220709}{5329} a + \frac{35556229}{10658} : -\frac{1698743871}{1556068} a^{2} + \frac{898050317}{1556068} a + \frac{1804012198}{389017} : 1\right)$
Height \(3.24105650937246\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-784 a^{2} + 416 a + 3333 : -1013 a^{2} + 535 a + 4302 : 1\right)$
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: \( 3.24105650937246 \)
Period: \( 18.7646538140429 \)
Tamagawa product: \( 4 \)
Torsion order: \(4\)
Leading coefficient: \( 2.56593045791911 \)
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\((j)_{-}\)
\((-a+1)\) \(2\) \(4\) \(I_9^{*}\) Additive \(1\) \(6\) \(19\) \(1\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 64.7-d consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.