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([-1,0,1,0]),K([-1,0,0,0]),K([1,-3,0,1]),K([2383,10834,-3604,-4355]),K([565564,2571802,-854686,-1033349])])
gp: E = ellinit([Polrev([-1,0,1,0]),Polrev([-1,0,0,0]),Polrev([1,-3,0,1]),Polrev([2383,10834,-3604,-4355]),Polrev([565564,2571802,-854686,-1033349])], K);
magma: E := EllipticCurve([K![-1,0,1,0],K![-1,0,0,0],K![1,-3,0,1],K![2383,10834,-3604,-4355],K![565564,2571802,-854686,-1033349]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^3+2a^2+6a-4)\) | = | \((-a^3+a^2+3a-2)\cdot(2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 144 \) | = | \(9\cdot16\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-248832)\) | = | \((-a^3+a^2+3a-2)^{10}\cdot(2)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 3833759992447475122176 \) | = | \(9^{10}\cdot16^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{19465109}{248832} \) | ||
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: | \(1\) |
| Generator | $\left(80 a^{3} + 67 a^{2} - 196 a - 41 : 1128 a^{3} + 937 a^{2} - 2800 a - 618 : 1\right)$ |
| Height | \(0.68640497937270242268490890046288388447\) |
| 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: | $\left(49 a^{3} + 41 a^{2} - 120 a - 25 : -59 a^{3} - 49 a^{2} + 146 a + 32 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.68640497937270242268490890046288388447 \) | ||
| Period: | \( 3.1399303719779948954928250029760338940 \) | ||
| Tamagawa product: | \( 20 \) = \(2\cdot( 2 \cdot 5 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.28515105616729 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+3a-2)\) | \(9\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
| \((2)\) | \(16\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
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 |
| \(5\) | 5B.1.4[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
144.1-b
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 4 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 450.f3 |
| \(\Q\) | 450.b3 |
| \(\Q(\sqrt{5}) \) | 2.2.5.1-36.1-a2 |
| \(\Q(\sqrt{5}) \) | a curve with conductor norm 8100 (not in the database) |