Base field 6.6.434581.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,3,1,-5,-1,1]),K([2,-5,-8,7,4,-2]),K([-2,5,12,-10,-6,3]),K([-3,-2,6,5,0,-1]),K([-3,-8,1,18,4,-4])])
gp: E = ellinit([Polrev([0,3,1,-5,-1,1]),Polrev([2,-5,-8,7,4,-2]),Polrev([-2,5,12,-10,-6,3]),Polrev([-3,-2,6,5,0,-1]),Polrev([-3,-8,1,18,4,-4])], K);
magma: E := EllipticCurve([K![0,3,1,-5,-1,1],K![2,-5,-8,7,4,-2],K![-2,5,12,-10,-6,3],K![-3,-2,6,5,0,-1],K![-3,-8,1,18,4,-4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3a^5-7a^4-9a^3+17a^2+3a-5)\) | = | \((3a^5-7a^4-9a^3+17a^2+3a-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^5+7a^4+18a^3-21a^2-13a+10)\) | = | \((3a^5-7a^4-9a^3+17a^2+3a-5)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -24389 \) | = | \(-29^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2227643651}{24389} a^{5} - \frac{6600751439}{24389} a^{4} - \frac{5119963945}{24389} a^{3} + \frac{20357393301}{24389} a^{2} + \frac{411157352}{24389} a - \frac{13119189482}{24389} \) | ||
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 | $\left(a^{5} - 2 a^{4} - 4 a^{3} + 4 a^{2} + 5 a + 1 : -7 a^{5} + 9 a^{4} + 31 a^{3} - 8 a^{2} - 21 a - 4 : 1\right)$ |
| Height | \(0.00077652884849094157253856608855361059583\) |
| 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: | \( 0.00077652884849094157253856608855361059583 \) | ||
| Period: | \( 95834.381593173948413872543324347624020 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.03196 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3a^5-7a^4-9a^3+17a^2+3a-5)\) | \(29\) | \(3\) | \(I_{3}\) | 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 |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
29.2-d
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.