Properties

Label 4.4.17428.1-12.2-c2
Base field 4.4.17428.1
Conductor norm \( 12 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.17428.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + 4 x + 6 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([6, 4, -6, -1, 1]))
 
gp: K = nfinit(Polrev([6, 4, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, 4, -6, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-36a^{3}+75a^{2}+172a-306\right){x}-358a^{3}+661a^{2}+1652a-2788\)
sage: E = EllipticCurve([K([-3,-4,1,1]),K([2,-3,-1,1]),K([-2,0,1,0]),K([-306,172,75,-36]),K([-2788,1652,661,-358])])
 
gp: E = ellinit([Polrev([-3,-4,1,1]),Polrev([2,-3,-1,1]),Polrev([-2,0,1,0]),Polrev([-306,172,75,-36]),Polrev([-2788,1652,661,-358])], K);
 
magma: E := EllipticCurve([K![-3,-4,1,1],K![2,-3,-1,1],K![-2,0,1,0],K![-306,172,75,-36],K![-2788,1652,661,-358]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^3+3a^2-5a-6)\) = \((a^3+a^2-3a-2)\cdot(-a^2+2a+1)\cdot(a^2-a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 12 \) = \(2\cdot2\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((17a^3-33a^2-70a+138)\) = \((a^3+a^2-3a-2)^{2}\cdot(-a^2+2a+1)^{3}\cdot(a^2-a-3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 629856 \) = \(2^{2}\cdot2^{3}\cdot3^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1264686676441}{157464} a^{3} + \frac{226053647473}{19683} a^{2} - \frac{196923906803}{8748} a - \frac{1648890452299}{78732} \)
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: $\left(\frac{7}{4} a^{3} - 4 a^{2} - \frac{19}{2} a + \frac{35}{2} : \frac{7}{8} a^{3} + \frac{3}{2} a^{2} - \frac{7}{4} a - \frac{17}{4} : 1\right)$
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: \( 318.59796092231442613571744676431711828 \)
Tamagawa product: \( 6 \)  =  \(2\cdot3\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 3.62001587731500 \)
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\((j)_{-}\)
\((a^3+a^2-3a-2)\) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-a^2+2a+1)\) \(2\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a^2-a-3)\) \(3\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 12.2-c consists of curves linked by isogenies of degrees dividing 6.

Base change

This elliptic curve is not a \(\Q\)-curve.

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