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([-6,4/3,5/3,-1/3]),K([1,-4/3,-2/3,1/3]),K([-264,346/3,230/3,-79/3]),K([2170,-954,-633,218])])
gp: E = ellinit([Polrev([1,-4/3,-2/3,1/3]),Polrev([-6,4/3,5/3,-1/3]),Polrev([1,-4/3,-2/3,1/3]),Polrev([-264,346/3,230/3,-79/3]),Polrev([2170,-954,-633,218])], K);
magma: E := EllipticCurve([K![1,-4/3,-2/3,1/3],K![-6,4/3,5/3,-1/3],K![1,-4/3,-2/3,1/3],K![-264,346/3,230/3,-79/3],K![2170,-954,-633,218]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((1/3a^3-2/3a^2-4/3a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(4^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((16)\) | = | \((1/3a^3-2/3a^2-4/3a+1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -65536 \) | = | \(-4^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -141475015320 a^{3} + 65462998280 a^{2} + 1090960239300 a + 828263760716 \) | ||
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: | \(0\) |
| 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: | $\left(a^{3} - 3 a^{2} - 4 a + \frac{19}{2} : -\frac{3}{4} a^{3} + 2 a^{2} + \frac{7}{2} a - \frac{27}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 261.32363415891750112731123451655610112 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.21747552592921 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/3a^3-2/3a^2-4/3a+1)\) | \(4\) | \(1\) | \(IV^{*}\) | Additive | \(1\) | \(2\) | \(8\) | \(0\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.