Properties

Label 4.4.1125.1-145.2-d2
Base field \(\Q(\zeta_{15})^+\)
Conductor norm \( 145 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{15})^+\)

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

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

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(-454716a^{3}-376136a^{2}+1131604a+248835\right){x}-229726834a^{3}-190005324a^{2}+571750203a+125733493\)
sage: E = EllipticCurve([K([-2,-3,1,1]),K([-2,1,1,0]),K([0,0,0,0]),K([248835,1131604,-376136,-454716]),K([125733493,571750203,-190005324,-229726834])])
 
gp: E = ellinit([Polrev([-2,-3,1,1]),Polrev([-2,1,1,0]),Polrev([0,0,0,0]),Polrev([248835,1131604,-376136,-454716]),Polrev([125733493,571750203,-190005324,-229726834])], K);
 
magma: E := EllipticCurve([K![-2,-3,1,1],K![-2,1,1,0],K![0,0,0,0],K![248835,1131604,-376136,-454716],K![125733493,571750203,-190005324,-229726834]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4a^3+a^2-12a+1)\) = \((-a-1)\cdot(a^2-a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 145 \) = \(5\cdot29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((124a^3+407a^2-450a-938)\) = \((-a-1)\cdot(a^2-a-3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 86249381545 \) = \(5\cdot29^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{84345529948654163101422494794}{86249381545} a^{3} + \frac{69761421738141200017547121122}{86249381545} a^{2} - \frac{209921659840910908051054641353}{86249381545} a - \frac{46163838435148922680899820821}{86249381545} \)
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: $\left(-196 a^{3} - 166 a^{2} + 482 a + 112 : 201 a^{3} + 163 a^{2} - 505 a - 107 : 1\right)$
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: \( 0.41779101632813874906306008859170076557 \)
Tamagawa product: \( 7 \)  =  \(1\cdot7\)
Torsion order: \(2\)
Leading coefficient: \( 1.06811241908055 \)
Analytic order of Ш: \( 49 \) (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\((j)_{-}\)
\((-a-1)\) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((a^2-a-3)\) \(29\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

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
\(7\) 7B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 145.2-d consists of curves linked by isogenies of degrees dividing 14.

Base change

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

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