Base field \(\Q(\zeta_{21})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -8, 8, 6, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -8, 8, 6, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 8, 6, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,6,2,-5,0,1]),K([2,-1,-4,4,1,-1]),K([-4,2,2,-4,0,1]),K([1,8,-4,-11,2,3]),K([-2,19,-14,-24,6,6])])
gp: E = ellinit([Polrev([-3,6,2,-5,0,1]),Polrev([2,-1,-4,4,1,-1]),Polrev([-4,2,2,-4,0,1]),Polrev([1,8,-4,-11,2,3]),Polrev([-2,19,-14,-24,6,6])], K);
magma: E := EllipticCurve([K![-3,6,2,-5,0,1],K![2,-1,-4,4,1,-1],K![-4,2,2,-4,0,1],K![1,8,-4,-11,2,3],K![-2,19,-14,-24,6,6]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+a^4+6a^3-5a^2-8a+3)\) | = | \((-a^5+a^4+6a^3-5a^2-8a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^5-5a^3+2a^2+5a-5)\) | = | \((-a^5+a^4+6a^3-5a^2-8a+3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -41 \) | = | \(-41\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{2317236930}{41} a^{5} + \frac{6745914900}{41} a^{4} + \frac{942380973}{41} a^{3} - \frac{15400001988}{41} a^{2} + \frac{10454504706}{41} a - \frac{1224790551}{41} \) | ||
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 | $\left(-a^{4} + a^{3} + 3 a^{2} - 5 a + 2 : a^{4} + 3 a^{3} - 5 a^{2} - 9 a + 9 : 1\right)$ |
| Height | \(0.0041762855719201394771630882797172046147\) |
| 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: | \( 0.0041762855719201394771630882797172046147 \) | ||
| Period: | \( 59892.466146077458345329072565085321460 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.22786 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+a^4+6a^3-5a^2-8a+3)\) | \(41\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
41.4-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.