Base field \(\Q(\zeta_{20})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([5, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0]),K([2,0,-1,0]),K([-3,0,1,0]),K([9,0,-10,0]),K([-42,0,31,0])])
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([2,0,-1,0]),Polrev([-3,0,1,0]),Polrev([9,0,-10,0]),Polrev([-42,0,31,0])], K);
magma: E := EllipticCurve([K![-3,0,1,0],K![2,0,-1,0],K![-3,0,1,0],K![9,0,-10,0],K![-42,0,31,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3a^3-3a^2-9a+6)\) | = | \((a^3-a^2-3a+2)\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 324 \) | = | \(4\cdot81\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((18)\) | = | \((a^3-a^2-3a+2)^{2}\cdot(3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 104976 \) | = | \(4^{2}\cdot81^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{131872229}{18} \) | ||
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 | $\left(a^{2} - a - 2 : -2 a^{2} - 2 a + 2 : 1\right)$ |
| Height | \(0.26021102422771097657212502052091857935\) |
| Torsion structure: | \(\Z/10\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(2 a^{2} - 4 : a^{2} - 4 : 1\right)$ |
| 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: | \( 0.26021102422771097657212502052091857935 \) | ||
| Period: | \( 1962.4564824862468096830156268600211838 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(10\) | ||
| Leading coefficient: | \( 1.82696703838506 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-3a+2)\) | \(4\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((3)\) | \(81\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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.1.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
324.1-d
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 |
|---|---|
| \(\Q\) | 1200.g2 |
| \(\Q\) | 1200.m2 |
| \(\Q(\sqrt{5}) \) | a curve with conductor norm 57600 (not in the database) |
| \(\Q(\sqrt{5}) \) | 2.2.5.1-36.1-a4 |