Base field 3.3.993.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -6, -1, 1]))
gp: K = nfinit(Polrev([3, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,0,1]),K([-3,-1,1]),K([-4,1,1]),K([-13,19,-10]),K([38,-92,15])])
gp: E = ellinit([Polrev([-4,0,1]),Polrev([-3,-1,1]),Polrev([-4,1,1]),Polrev([-13,19,-10]),Polrev([38,-92,15])], K);
magma: E := EllipticCurve([K![-4,0,1],K![-3,-1,1],K![-4,1,1],K![-13,19,-10],K![38,-92,15]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a-3)\) | = | \((-a)\cdot(a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-190a^2+267a-384)\) | = | \((-a)^{17}\cdot(a-2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 645700815 \) | = | \(3^{17}\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{61857233}{98415} a^{2} + \frac{337862188}{98415} a - \frac{400078184}{32805} \) | ||
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 | $\left(-a^{2} + 1 : 3 a^{2} - 5 a + 4 : 1\right)$ |
| Height | \(0.043344838717241898807757021324021882191\) |
| 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: | \( 0.043344838717241898807757021324021882191 \) | ||
| Period: | \( 31.980581837780066543453030967926650049 \) | ||
| Tamagawa product: | \( 17 \) = \(17\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.24346503 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(17\) | \(I_{17}\) | Split multiplicative | \(-1\) | \(1\) | \(17\) | \(17\) |
| \((a-2)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 15.1-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.