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
sage: E = EllipticCurve([K([1,0,0,0]),K([1,0,0,0]),K([1,0,0,0]),K([-10,0,0,0]),K([-10,0,0,0])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([1,0,0,0]),Polrev([1,0,0,0]),Polrev([-10,0,0,0]),Polrev([-10,0,0,0])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![1,0,0,0],K![1,0,0,0],K![-10,0,0,0],K![-10,0,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-5a+1)\) | = | \((-a-1)\cdot(-a^3+a^2+3a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 45 \) | = | \(5\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((50625)\) | = | \((-a-1)^{16}\cdot(-a^3+a^2+3a-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6568408355712890625 \) | = | \(5^{16}\cdot9^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{111284641}{50625} \) | ||
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: | \(\Z/2\Z\oplus\Z/8\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-1 : 0 : 1\right)$ | $\left(15 a^{3} - 45 a + 23 : 75 a^{3} - 225 a + 123 : 1\right)$ |
| 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: | \( 61.571572328209780501083110512035939105 \) | ||
| Tamagawa product: | \( 128 \) = \(2^{4}\cdot2^{3}\) | ||
| Torsion order: | \(16\) | ||
| Leading coefficient: | \( 0.917854808049480 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a-1)\) | \(5\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
| \((-a^3+a^2+3a-2)\) | \(9\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
45.1-b
consists of curves linked by isogenies of
degrees dividing 32.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 4 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 15.a5 |
| \(\Q\) | 75.b5 |
| \(\Q(\sqrt{5}) \) | 2.2.5.1-45.1-a5 |
| \(\Q(\sqrt{5}) \) | 2.2.5.1-2025.1-b5 |