Properties

Label 4.4.2304.1-72.1-b4
Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Conductor \((-a^3+a^2+5a-5)\)
Conductor norm \( 72 \)
CM no
Base change yes: 24.a4,144.b4,192.d4,576.b4
Q-curve yes
Torsion order \( 16 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}, \sqrt{3})\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -4, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, 0, -4, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 0, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+a^2+5a-5)\) = \((a^3-4a+1)^{3}\cdot(a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 72 \) = \(2^{3}\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((36)\) = \((a^3-4a+1)^{8}\cdot(a^2-2)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1679616 \) = \(2^{8}\cdot9^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{35152}{9} \)
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 $\left(a + 1 : -a^{3} + 4 a + 1 : 1\right)$
Height \(0.269818466169295\)
Torsion structure: \(\Z/2\Z\times\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(2 a^{3} - a^{2} - 7 a + 4 : 3 a^{3} - 2 a^{2} - 12 a + 7 : 1\right)$ $\left(\frac{1}{2} : -\frac{3}{4} a^{3} + \frac{9}{4} a : 1\right)$
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: \( 0.269818466169295 \)
Period: \( 1384.17317605613 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2^{2}\)
Torsion order: \(16\)
Leading coefficient: \( 1.94518480872993 \)
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\((j)_{-}\)
\((a^3-4a+1)\) \(2\) \(4\) \(I_1^{*}\) Additive \(1\) \(3\) \(8\) \(0\)
\((a^2-2)\) \(9\) \(4\) \(I_{4}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 72.1-b consists of curves linked by isogenies of degrees dividing 32.

Base change

This curve is the base change of elliptic curves 24.a4, 144.b4, 192.d4, 576.b4, defined over \(\Q\), so it is also a \(\Q\)-curve.