Base field \(\Q(\sqrt{-7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
gp: K = nfinit(Polrev([2, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,0]),K([1,1]),K([-22,-30]),K([-81,-9])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([1,1]),Polrev([-22,-30]),Polrev([-81,-9])], K);
magma: E := EllipticCurve([K![1,1],K![-1,0],K![1,1],K![-22,-30],K![-81,-9]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((154a-110)\) | = | \((a)\cdot(-a+1)^{4}\cdot(-2a+3)^{2}\cdot(2a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 42592 \) | = | \(2\cdot2^{4}\cdot11^{2}\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4088832a-13629440)\) | = | \((a)^{13}\cdot(-a+1)^{10}\cdot(-2a+3)^{4}\cdot(2a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 163470238547968 \) | = | \(2^{13}\cdot2^{10}\cdot11^{4}\cdot11^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{3287573457}{10903552} a + \frac{14322326347}{10903552} \) | ||
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 | $\left(-a + 7 : 9 a - 21 : 1\right)$ | $\left(-9 a + 3 : 29 a + 11 : 1\right)$ |
| Heights | \(0.16022243033527536880515206282131272112\) | \(0.072920469427481665964055773519310193922\) |
| 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: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.0073804284219367960820888724170918803936 \) | ||
| Period: | \( 0.90042207243989892816217898660190982025 \) | ||
| Tamagawa product: | \( 468 \) = \(13\cdot2^{2}\cdot3\cdot3\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 9.4040412448902778591418385819849411838 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(13\) | \(I_{13}\) | Split multiplicative | \(-1\) | \(1\) | \(13\) | \(13\) |
| \((-a+1)\) | \(2\) | \(4\) | \(I_{2}^{*}\) | Additive | \(1\) | \(4\) | \(10\) | \(0\) |
| \((-2a+3)\) | \(11\) | \(3\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
| \((2a+1)\) | \(11\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 42592.18-g 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.