Base field \(\Q(\zeta_{7})^+\)
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -1, 1]))
gp: K = nfinit(Polrev([1, -2, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,1,1]),K([-3,0,1]),K([-1,0,1]),K([-27,-9,14]),K([-74,93,-28])])
gp: E = ellinit([Polrev([-1,1,1]),Polrev([-3,0,1]),Polrev([-1,0,1]),Polrev([-27,-9,14]),Polrev([-74,93,-28])], K);
magma: E := EllipticCurve([K![-1,1,1],K![-3,0,1],K![-1,0,1],K![-27,-9,14],K![-74,93,-28]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^2+2a-6)\) | = | \((2)\cdot(a^2+a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 104 \) | = | \(8\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((9896a^2-20056a-7432)\) | = | \((2)^{3}\cdot(a^2+a-3)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -5429503678976 \) | = | \(-8^{3}\cdot13^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{5104794835848393}{42417997492} a^{2} - \frac{5647619780915441}{84835994984} a - \frac{5729473372444955}{21208998746} \) | ||
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: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-3 a^{2} - 2 a + 12 : 15 a^{2} - 17 a - 16 : 1\right)$ |
| 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: | \( 4.6827996322662158290378578085291613168 \) | ||
| Tamagawa product: | \( 9 \) = \(1\cdot3^{2}\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 0.66897137603803083271969397264702304525 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(8\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((a^2+a-3)\) | \(13\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
104.1-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.