Properties

Label 4.4.4752.1-33.1-d3
Base field 4.4.4752.1
Conductor norm \( 33 \)
CM no
Base change yes
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+\left(a^{2}-a-1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-77a^{2}+77a-124\right){x}+719a^{2}-719a+358\)
sage: E = EllipticCurve([K([-1,-1,1,0]),K([3,1,-1,0]),K([1,0,0,0]),K([-124,77,-77,0]),K([358,-719,719,0])])
 
gp: E = ellinit([Polrev([-1,-1,1,0]),Polrev([3,1,-1,0]),Polrev([1,0,0,0]),Polrev([-124,77,-77,0]),Polrev([358,-719,719,0])], K);
 
magma: E := EllipticCurve([K![-1,-1,1,0],K![3,1,-1,0],K![1,0,0,0],K![-124,77,-77,0],K![358,-719,719,0]]);
 

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: \((170299a^2-170299a-436478)\) = \((a^3-a^2-4a+1)^{2}\cdot(a^2-2a-2)^{20}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6054749954393040082809 \) = \(3^{2}\cdot11^{20}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1081911102879025664}{77812273803} a^{2} - \frac{1081911102879025664}{77812273803} a - \frac{3980255358140970688}{77812273803} \)
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: $\left(\frac{15}{2} a^{2} - \frac{15}{2} a - 17 : -\frac{11}{4} a^{2} + \frac{11}{4} a - \frac{21}{4} : 1\right)$
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: \( 77.218052178530693302460715647477762643 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.12016146005862 \)
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\((j)_{-}\)
\((a^3-a^2-4a+1)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a^2-2a-2)\) \(11\) \(2\) \(I_{20}\) Non-split multiplicative \(1\) \(1\) \(20\) \(20\)

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
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 33.1-d consists of curves linked by isogenies of degrees dividing 10.

Base change

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

Base field Curve
\(\Q(\sqrt{3}) \) 2.2.12.1-33.2-a3
\(\Q(\sqrt{3}) \) 2.2.12.1-1089.2-f3