Base field 4.4.13888.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 6 x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 6, -7, -2, 1]))
gp: K = nfinit(Polrev([9, 6, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 6, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-4/3,-2/3,1/3]),K([-1,1/3,2/3,-1/3]),K([2,-4/3,-2/3,1/3]),K([-19,-61/3,1/3,7/3]),K([27,115/3,8/3,-16/3])])
gp: E = ellinit([Polrev([1,-4/3,-2/3,1/3]),Polrev([-1,1/3,2/3,-1/3]),Polrev([2,-4/3,-2/3,1/3]),Polrev([-19,-61/3,1/3,7/3]),Polrev([27,115/3,8/3,-16/3])], K);
magma: E := EllipticCurve([K![1,-4/3,-2/3,1/3],K![-1,1/3,2/3,-1/3],K![2,-4/3,-2/3,1/3],K![-19,-61/3,1/3,7/3],K![27,115/3,8/3,-16/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/3a^3+1/3a^2-1/3a-1)\) | = | \((1/3a^3+1/3a^2-1/3a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 7 \) | = | \(7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4/3a^3-5/3a^2-28/3a-4)\) | = | \((1/3a^3+1/3a^2-1/3a-1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -343 \) | = | \(-7^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{7848998400}{49} a^{3} + \frac{4430760448}{7} a^{2} - \frac{797645568}{7} a - \frac{36197779648}{49} \) | ||
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 | $\left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + \frac{7}{3} a + 1 : -\frac{2}{3} a^{3} + \frac{4}{3} a^{2} + \frac{5}{3} a - 1 : 1\right)$ |
| Height | \(0.47881605862038052750581055982624926738\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a + \frac{1}{2} : -\frac{1}{4} a^{3} + \frac{3}{2} a + \frac{1}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.47881605862038052750581055982624926738 \) | ||
| Period: | \( 222.04088812848876321320285606048320745 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.70646904345154 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/3a^3+1/3a^2-1/3a-1)\) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3Ns |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
7.2-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.