Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-145.3-a1
Conductor \((145,-a^{3} - a^{2} + 3 a - 2)\)
Conductor norm \( 145 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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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: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 4*x^2 + 4*x + 1)
 
gp: K = nfinit(a^4 - a^3 - 4*a^2 + 4*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) x y + \left(a^{3} - 3 a + 1\right) y = x^{3} + \left(-a^{3} + a^{2} + 4 a - 2\right) x^{2} + \left(767 a^{3} + 261 a^{2} - 2731 a - 600\right) x + 14439 a^{3} + 4899 a^{2} - 51204 a - 10817 \)
sage: E = EllipticCurve(K, [a + 1, -a^3 + a^2 + 4*a - 2, a^3 - 3*a + 1, 767*a^3 + 261*a^2 - 2731*a - 600, 14439*a^3 + 4899*a^2 - 51204*a - 10817])
 
gp: E = ellinit([a + 1, -a^3 + a^2 + 4*a - 2, a^3 - 3*a + 1, 767*a^3 + 261*a^2 - 2731*a - 600, 14439*a^3 + 4899*a^2 - 51204*a - 10817],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, -a^3 + a^2 + 4*a - 2, a^3 - 3*a + 1, 767*a^3 + 261*a^2 - 2731*a - 600, 14439*a^3 + 4899*a^2 - 51204*a - 10817]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((145,-a^{3} - a^{2} + 3 a - 2)\) = \( \left(-a - 1\right) \cdot \left(-a^{3} + a^{2} + 4 a - 2\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 145 \) = \( 5 \cdot 29 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((145,a^{3} - 3 a + 53,a + 131,a^{2} + 94)\) = \( \left(-a - 1\right) \cdot \left(-a^{3} + a^{2} + 4 a - 2\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 145 \) = \( 5 \cdot 29 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{765623948004282479}{145} a^{3} - \frac{258980786426845107}{145} a^{2} + \frac{2715911939787230208}{145} a + \frac{572103573083015331}{145} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(11 a^{3} + \frac{19}{4} a^{2} - \frac{77}{2} a - \frac{41}{4} : -\frac{111}{8} a^{3} - \frac{41}{8} a^{2} + \frac{383}{8} a + \frac{81}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

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\((j)_{-}\)
\( \left(-a - 1\right) \) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(-a^{3} + a^{2} + 4 a - 2\right) \) \(29\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 145.3-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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