Properties

Label 4.4.4752.1-33.1-g1
Base field 4.4.4752.1
Conductor norm \( 33 \)
CM no
Base change yes
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 4.4.4752.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -3, -2, 1]))
 
gp: K = nfinit(Polrev([1, 4, -3, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -3, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(6a^{3}+4a^{2}-10a-4\right){x}-123a^{3}-61a^{2}+217a+49\)
sage: E = EllipticCurve([K([2,-3,-1,1]),K([-2,0,1,0]),K([1,0,0,0]),K([-4,-10,4,6]),K([49,217,-61,-123])])
 
gp: E = ellinit([Polrev([2,-3,-1,1]),Polrev([-2,0,1,0]),Polrev([1,0,0,0]),Polrev([-4,-10,4,6]),Polrev([49,217,-61,-123])], K);
 
magma: E := EllipticCurve([K![2,-3,-1,1],K![-2,0,1,0],K![1,0,0,0],K![-4,-10,4,6],K![49,217,-61,-123]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a+1)\) = \((a^3-a^2-4a+1)\cdot(a^2-2a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 33 \) = \(3\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-6a^2+6a+15)\) = \((a^3-a^2-4a+1)^{4}\cdot(a^2-2a-2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 9801 \) = \(3^{4}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{3335680}{33} a^{2} + \frac{3335680}{33} a + \frac{12449728}{33} \)
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 $\left(a^{3} - 2 a + 1 : a^{3} + 2 a^{2} - a - 4 : 1\right)$
Height \(0.21310074772669094768771726274094793204\)
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: $\left(\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - a - \frac{1}{2} : -a^{3} - \frac{3}{4} a^{2} + \frac{7}{4} a + \frac{1}{4} : 1\right)$
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: \( 0.21310074772669094768771726274094793204 \)
Period: \( 155.51049578617924245383960476671322025 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.92294319402219 \)
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\((j)_{-}\)
\((a^3-a^2-4a+1)\) \(3\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((a^2-2a-2)\) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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.
Its isogeny class 33.1-g consists of curves linked by isogenies of degree 2.

Base change

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

Base field Curve
\(\Q(\sqrt{3}) \) 2.2.12.1-363.2-d1
\(\Q(\sqrt{3}) \) 2.2.12.1-99.1-a1