Properties

Label 4.4.1125.1-145.1-d1
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

\({y}^2+\left(a^{3}+a^{2}-2a-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+{x}^{2}+\left(-70933a^{3}-60234a^{2}+173150a+38181\right){x}-15136874a^{3}-12573990a^{2}+37555267a+8262428\)
sage: E = EllipticCurve([K([-1,-2,1,1]),K([1,0,0,0]),K([0,-2,0,1]),K([38181,173150,-60234,-70933]),K([8262428,37555267,-12573990,-15136874])])
 
gp: E = ellinit([Polrev([-1,-2,1,1]),Polrev([1,0,0,0]),Polrev([0,-2,0,1]),Polrev([38181,173150,-60234,-70933]),Polrev([8262428,37555267,-12573990,-15136874])], K);
 
magma: E := EllipticCurve([K![-1,-2,1,1],K![1,0,0,0],K![0,-2,0,1],K![38181,173150,-60234,-70933],K![8262428,37555267,-12573990,-15136874]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4a^3-13a+1)\) = \((-a-1)\cdot(-a^3-a^2+2a+3)\)
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: \((46a^3-78a^2-545a+404)\) = \((-a-1)\cdot(-a^3-a^2+2a+3)^{7}\)
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: \( -\frac{127460459953705744354635337823}{86249381545} a^{3} - \frac{43114930005051581253212843029}{86249381545} a^{2} + \frac{452142801599258433081453134591}{86249381545} a + \frac{95243335102582476759197612687}{86249381545} \)
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: $\left(182 a^{3} + 141 a^{2} - 475 a - 105 : -392 a^{3} - 321 a^{2} + 981 a + 215 : 1\right)$
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 \(-1\) \(1\) \(1\) \(1\)
\((-a^3-a^2+2a+3)\) \(29\) \(7\) \(I_{7}\) 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 145.1-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.