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([-1,0]),K([1,1]),K([-229718,52701]),K([76019186,-17439999])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,1]),Polrev([-229718,52701]),Polrev([76019186,-17439999])], K);
magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,1],K![-229718,52701],K![76019186,-17439999]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a+10)\) | = | \((-3a+13)^{3}\cdot(-a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 24 \) | = | \(2^{3}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3328a-16940)\) | = | \((-3a+13)^{4}\cdot(-a+4)^{14}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 76527504 \) | = | \(2^{4}\cdot3^{14}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{2379806720}{4782969} a - \frac{24019195904}{4782969} \) | ||
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 | $\left(-\frac{275}{9} a + \frac{1202}{9} : \frac{21532}{27} a - \frac{93928}{27} : 1\right)$ |
| Height | \(0.62768755281757204557634996536436870915\) |
| 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: | \( 0.62768755281757204557634996536436870915 \) | ||
| Period: | \( 5.4239831668978780497966596279989756632 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.1242446908285979973836057355524308332 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-3a+13)\) | \(2\) | \(2\) | \(III\) | Additive | \(-1\) | \(3\) | \(4\) | \(0\) |
| \((-a+4)\) | \(3\) | \(2\) | \(I_{14}\) | Non-split multiplicative | \(1\) | \(1\) | \(14\) | \(14\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 24.2-c consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.