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([1,1]),K([0,1]),K([0,0]),K([-120392,27630]),K([32632300,-7486350])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([0,0]),Polrev([-120392,27630]),Polrev([32632300,-7486350])], K);
magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![-120392,27630],K![32632300,-7486350]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((26a+114)\) | = | \((-3a+13)^{3}\cdot(a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 152 \) | = | \(2^{3}\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-608)\) | = | \((-3a+13)^{10}\cdot(a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 369664 \) | = | \(2^{10}\cdot19^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{31250}{19} \) | ||
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: | \(2\) | |
| Generators | $\left(-\frac{19}{2} a + \frac{601}{16} : -\frac{25541}{32} a + \frac{223321}{64} : 1\right)$ | $\left(6 a - 30 : -1005 a + 4391 : 1\right)$ |
| Heights | \(2.6238351942172237621195884469109995813\) | \(0.10336644109328545272016429089996029955\) |
| 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: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.10026256773287743117512272609112154118 \) | ||
| Period: | \( 22.710380601437086575438799830381941846 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 4.1790389780249137638189870268142753646 \) | ||
| 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 | \(1\) | \(3\) | \(10\) | \(0\) |
| \((a)\) | \(19\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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 152.1-c consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 304.b1 |
| \(\Q\) | 2888.b1 |