Base field \(\Q(\zeta_{9})^+\)
Generator \(a\), with minimal polynomial \( x^{3} - 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 0, 1]))
gp: K = nfinit(Polrev([-1, -3, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0]),K([1,1,-1]),K([-2,0,1]),K([168,412,-273]),K([1822,3940,-3801])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([1,1,-1]),Polrev([-2,0,1]),Polrev([168,412,-273]),Polrev([1822,3940,-3801])], K);
magma: E := EllipticCurve([K![1,1,0],K![1,1,-1],K![-2,0,1],K![168,412,-273],K![1822,3940,-3801]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-3a-4)\) | = | \((-a^2+1)\cdot(a^2+a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 51 \) | = | \(3\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((28206a^2-4329a-48420)\) | = | \((-a^2+1)^{8}\cdot(a^2+a-3)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -45767944570401 \) | = | \(-3^{8}\cdot17^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{15897092117456459660800496}{62781816969} a^{2} - \frac{73067274470120883503414977}{188345450907} a - \frac{31128269893684578702277955}{188345450907} \) | ||
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: | $\left(-\frac{5}{4} a^{2} + \frac{21}{2} a + \frac{3}{4} : -\frac{41}{8} a^{2} - \frac{15}{4} a + \frac{5}{4} : 1\right)$ |
| 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: | \( 1.4764520048470304683879173308792195063 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot2^{3}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.65620089104312465261685214705743089170 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2+1)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((a^2+a-3)\) | \(17\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
51.3-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.