Base field 4.4.8069.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 5, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-4,1,1]),K([-1,-3,1,1]),K([-1,-3,1,1]),K([-5,-11,6,4]),K([-5,-18,9,7])])
gp: E = ellinit([Polrev([-1,-4,1,1]),Polrev([-1,-3,1,1]),Polrev([-1,-3,1,1]),Polrev([-5,-11,6,4]),Polrev([-5,-18,9,7])], K);
magma: E := EllipticCurve([K![-1,-4,1,1],K![-1,-3,1,1],K![-1,-3,1,1],K![-5,-11,6,4],K![-5,-18,9,7]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2+a-4)\) | = | \((a^2+a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3+a^2-3a-6)\) | = | \((a^2+a-4)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 361 \) | = | \(19^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{468732461}{361} a^{3} - \frac{107902627}{361} a^{2} + \frac{2209071420}{361} a + \frac{374364855}{361} \) | ||
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: | \(0 \le r \le 1\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a^{3} - 2 a^{2} - 7 a + 10 : -4 a^{3} + 5 a^{2} + 25 a - 30 : 1\right)$ |
| 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: | \(0 \le r \le 1\) | ||
| Regulator: | not available | ||
| Period: | \( 622.70598152227703014998759867729480546 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.40202085800390 \) | ||
| Analytic order of Ш: | not available | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2+a-4)\) | \(19\) | \(2\) | \(I_{2}\) | 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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
19.1-c
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.