Properties

Label 2.2.12.1-4225.1-a1
Base field \(\Q(\sqrt{3}) \)
Conductor \((65)\)
Conductor norm \( 4225 \)
CM no
Base change yes: 1040.f2,585.h2
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-3, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}-{x}^{2}+4{x}-1\)
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,0]),K([4,0]),K([-1,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([4,0])),Pol(Vecrev([-1,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,0],K![4,0],K![-1,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((65)\) = \((a+4)\cdot(a-4)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4225 \) = \(13\cdot13\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4225)\) = \((a+4)^{2}\cdot(a-4)^{2}\cdot(5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 17850625 \) = \(13^{2}\cdot13^{2}\cdot25^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{6967871}{4225} \)
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 $\left(\frac{4}{3} : \frac{7}{9} a : 1\right)$
Height \(0.819087883055301\)
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: $\left(\frac{1}{4} : -\frac{1}{8} a : 1\right)$
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: \( 0.819087883055301 \)
Period: \( 6.46445089395728 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 6.11408553918631 \)
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\((j)_{-}\)
\((a+4)\) \(13\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-4)\) \(13\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((5)\) \(25\) \(2\) \(I_{2}\) 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 4225.1-a consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of 1040.f2, 585.h2, defined over \(\Q\), so it is also a \(\Q\)-curve.