Properties

Label 4.4.2304.1-72.1-b6
Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Conductor \((-a^3+a^2+5a-5)\)
Conductor norm \( 72 \)
CM no
Base change yes: 24.a2,144.b2,192.d2,576.b2
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

\({y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(16a^{3}-40a^{2}+16a+1\right){x}-119a^{3}+249a^{2}-16a-42\)
sage: E = EllipticCurve([K([1,1,0,0]),K([-2,0,1,0]),K([-1,0,1,0]),K([1,16,-40,16]),K([-42,-16,249,-119])])
 
gp: E = ellinit([Pol(Vecrev([1,1,0,0])),Pol(Vecrev([-2,0,1,0])),Pol(Vecrev([-1,0,1,0])),Pol(Vecrev([1,16,-40,16])),Pol(Vecrev([-42,-16,249,-119]))], K);
 
magma: E := EllipticCurve([K![1,1,0,0],K![-2,0,1,0],K![-1,0,1,0],K![1,16,-40,16],K![-42,-16,249,-119]]);
 

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: \((6)\) = \((a^3-4a+1)^{4}\cdot(a^2-2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1296 \) = \(2^{4}\cdot9^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{28756228}{3} \)
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 $\left(-13 a^{3} + 26 a^{2} + a - 5 : -193 a^{3} + 372 a^{2} + 52 a - 100 : 1\right)$
Height \(0.539636932338590\)
Torsion structure: \(\Z/2\Z\oplus\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(2 a^{3} - 4 a^{2} - 3 a + 3 : a^{3} - a^{2} : 1\right)$ $\left(3 a^{2} - 8 a + 4 : 4 a^{3} - 17 a^{2} + 21 a - 7 : 1\right)$
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: \( 0.539636932338590 \)
Period: \( 2768.34635211226 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
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\((j)_{-}\)
\((a^3-4a+1)\) \(2\) \(2\) \(III\) Additive \(1\) \(3\) \(4\) \(0\)
\((a^2-2)\) \(9\) \(2\) \(I_{2}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
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 24.a2, 144.b2, 192.d2, 576.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.