Base field \(\Q(\sqrt{69}) \)
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([0,1]),K([0,0]),K([1,0]),K([40,-7]),K([-134,30])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([1,0]),Polrev([40,-7]),Polrev([-134,30])], K);
magma: E := EllipticCurve([K![0,1],K![0,0],K![1,0],K![40,-7],K![-134,30]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a+8)\) | = | \((2)\cdot(-a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 20 \) | = | \(4\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((24a-56)\) | = | \((2)^{3}\cdot(-a+4)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -8000 \) | = | \(-4^{3}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{117621}{500} a + \frac{1288531}{1000} \) | ||
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 | $\left(-\frac{22}{25} a + \frac{54}{25} : -\frac{362}{125} a + \frac{2184}{125} : 1\right)$ |
| Height | \(2.0467771002205433343757104874248951973\) |
| 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: | $\left(-a + 3 : 2 a - 6 : 1\right)$ |
| 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: | \( 2.0467771002205433343757104874248951973 \) | ||
| Period: | \( 17.072361712069358513926648495679674535 \) | ||
| Tamagawa product: | \( 3 \) = \(3\cdot1\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 2.8044541782241512960952381367345794932 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((-a+4)\) | \(5\) | \(1\) | \(I_{3}\) | Non-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 |
|---|---|
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
20.1-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.