Base field \(\Q(\sqrt{-19}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, -1, 1]))
gp: K = nfinit(Polrev([5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([0,0]),K([-2,-1]),K([1,-1])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([0,0]),Polrev([-2,-1]),Polrev([1,-1])], K);
magma: E := EllipticCurve([K![1,0],K![-1,1],K![0,0],K![-2,-1],K![1,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a-10)\) | = | \((2)\cdot(-a)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 100 \) | = | \(4\cdot5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-558a-10)\) | = | \((2)\cdot(-a)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1562500 \) | = | \(4\cdot5^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{16129}{50} a + \frac{10942}{25} \) | ||
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 | $\left(-a - 1 : a - 2 : 1\right)$ |
| Height | \(0.21570037167620097926995683056935301244\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.21570037167620097926995683056935301244 \) | ||
| Period: | \( 4.1931484684060160488214450469201260231 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.6599855969938814931587072824253524856 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((-a)\) | \(5\) | \(2\) | \(I_{2}^{*}\) | Additive | \(1\) | \(2\) | \(8\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
100.1-a
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.