Properties

Label 4.4.2304.1-9.1-a7
Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Conductor norm \( 9 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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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(Polrev([1, 0, -4, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-67a^{3}+35a^{2}+251a-129\right){x}+381a^{3}-197a^{2}-1422a+735\)
sage: E = EllipticCurve([K([1,1,0,0]),K([-3,4,1,-1]),K([0,-4,0,1]),K([-129,251,35,-67]),K([735,-1422,-197,381])])
 
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-3,4,1,-1]),Polrev([0,-4,0,1]),Polrev([-129,251,35,-67]),Polrev([735,-1422,-197,381])], K);
 
magma: E := EllipticCurve([K![1,1,0,0],K![-3,4,1,-1],K![0,-4,0,1],K![-129,251,35,-67],K![735,-1422,-197,381]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-2)\) = \((a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^2+2)\) = \((a^2-2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 9 \) = \(9\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{467951528}{3} a^{3} + \frac{2339757640}{3} a + 382083912 \)
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: $\left(4 a^{3} - \frac{9}{4} a^{2} - \frac{31}{2} a + \frac{31}{4} : -\frac{11}{8} a^{3} + \frac{7}{8} a^{2} + \frac{47}{8} a - \frac{15}{8} : 1\right)$
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: \( 285.83292677657224671119127374294579967 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 1.48871316029465 \)
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\((j)_{-}\)
\((a^2-2)\) \(9\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 5, 10 and 20.
Its isogeny class 9.1-a consists of curves linked by isogenies of degrees dividing 20.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.