Properties

Label 4.4.18097.1-12.2-a1
Base field 4.4.18097.1
Conductor \((1/2a^3+1/2a^2-5/2a)\)
Conductor norm \( 12 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.18097.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 6 x + 4 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 6, -7, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([4, 6, -7, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 6, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-1\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{5}{2}a+3\right){x}^{2}+\left(-8a^{3}-10a^{2}+28a+8\right){x}-46a^{3}-75a^{2}+133a+82\)
sage: E = EllipticCurve([K([-1,-5/2,1/2,1/2]),K([3,5/2,-1/2,-1/2]),K([-1,-5/2,1/2,1/2]),K([8,28,-10,-8]),K([82,133,-75,-46])])
 
gp: E = ellinit([Pol(Vecrev([-1,-5/2,1/2,1/2])),Pol(Vecrev([3,5/2,-1/2,-1/2])),Pol(Vecrev([-1,-5/2,1/2,1/2])),Pol(Vecrev([8,28,-10,-8])),Pol(Vecrev([82,133,-75,-46]))], K);
 
magma: E := EllipticCurve([K![-1,-5/2,1/2,1/2],K![3,5/2,-1/2,-1/2],K![-1,-5/2,1/2,1/2],K![8,28,-10,-8],K![82,133,-75,-46]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/2a^3+1/2a^2-5/2a)\) = \((1/2a^3-1/2a^2-5/2a+1)\cdot(-1/2a^3+1/2a^2+7/2a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 12 \) = \(3\cdot4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1/2a^3+1/2a^2-9/2a-4)\) = \((1/2a^3-1/2a^2-5/2a+1)^{2}\cdot(-1/2a^3+1/2a^2+7/2a-3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 144 \) = \(3^{2}\cdot4^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{5026151635}{36} a^{3} + \frac{2933210821}{6} a^{2} - \frac{2210604581}{9} a - \frac{2009202809}{9} \)
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: \(0\)
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: $\left(\frac{5}{8} a^{3} + \frac{13}{8} a^{2} - \frac{13}{8} a - 5 : -\frac{11}{8} a^{3} - \frac{5}{2} a^{2} + \frac{13}{4} a + \frac{17}{8} : 1\right)$
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: \( 271.118690997167 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.01537638662772 \)
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\((j)_{-}\)
\((1/2a^3-1/2a^2-5/2a+1)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-1/2a^3+1/2a^2+7/2a-3)\) \(4\) \(2\) \(I_{2}\) 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 12.2-a consists of curves linked by isogenies of degree 2.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.