Base field \(\Q(\sqrt{19}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, 0, 1]))
gp: K = nfinit(Polrev([-19, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([-70,16]),K([-323,73])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,1]),Polrev([-70,16]),Polrev([-323,73])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![-70,16],K![-323,73]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((5a+22)\) | = | \((-a+4)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-28a-125)\) | = | \((-a+4)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 729 \) | = | \(3^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -884736 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-19})/2]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $N(\mathrm{U}(1))$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-3 a + 15 : -16 a + 65 : 1\right)$ |
| Height | \(1.6249221992145115389629239352658245284\) |
| 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: | \( 1.6249221992145115389629239352658245284 \) | ||
| Period: | \( 2.3259727904040606865934613452061647197 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.7341649213947706050943975599806767212 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+4)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=19\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -19 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -19 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
19.
Its isogeny class
9.3-c
consists of curves linked by isogenies of
degree 19.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.