Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([221,0]),K([17042,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,0]),Polrev([221,0]),Polrev([17042,0])], K);
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![221,0],K![17042,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-442a+221)\) | = | \((-2a+1)\cdot(-4a+1)\cdot(4a-3)\cdot(17)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 146523 \) | = | \(3\cdot13\cdot13\cdot289\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-124806800313)\) | = | \((-2a+1)^{4}\cdot(-4a+1)^{8}\cdot(4a-3)^{8}\cdot(17)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 15576737404369056897969 \) | = | \(3^{4}\cdot13^{8}\cdot13^{8}\cdot289\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1193377118543}{124806800313} \) | ||
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: | \(2\) | |
| Generators | $\left(\frac{507}{49} a - \frac{421}{49} : -\frac{4056}{343} a - \frac{42300}{343} : 1\right)$ | $\left(9 a - 19 : 24 a + 114 : 1\right)$ |
| Heights | \(0.79497221757411274938237879958096469745\) | \(0.79497221757411274938237879958096469744\) |
| Torsion structure: | \(\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generator: | $\left(19 : -179 : 1\right)$ | |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.62937786796683701708095430019357230969 \) | ||
| Period: | \( 0.19846059548612306344929969943065189879 \) | ||
| Tamagawa product: | \( 128 \) = \(2\cdot2^{3}\cdot2^{3}\cdot1\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 4.6153549183811385437700085032459938787 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((-4a+1)\) | \(13\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((4a-3)\) | \(13\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((17)\) | \(289\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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
146523.2-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 663.a5 |
| \(\Q\) | 1989.e5 |