Base field \(\Q(\sqrt{-67}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 17 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([17, -1, 1]))
gp: K = nfinit(Polrev([17, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([1,0]),K([-14,-5]),K([20,0])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([1,0]),Polrev([-14,-5]),Polrev([20,0])], K);
magma: E := EllipticCurve([K![1,1],K![-1,1],K![1,0],K![-14,-5],K![20,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3a-3)\) | = | \((3)\cdot(a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 153 \) | = | \(9\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((8019a+58077)\) | = | \((3)^{5}\cdot(a-1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4931831529 \) | = | \(9^{5}\cdot17^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{232667875}{6765201} a + \frac{58385000}{1193859} \) | ||
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 | $\left(0 : -5 : 1\right)$ |
| Height | \(0.17825229761437386555172391784569474010\) |
| 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: | \( 0.17825229761437386555172391784569474010 \) | ||
| Period: | \( 2.1715324885503777343598008324575622866 \) | ||
| Tamagawa product: | \( 20 \) = \(5\cdot2^{2}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.7831542875806121316651208272925358402 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3)\) | \(9\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((a-1)\) | \(17\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 153.2-a 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.