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([1,5,0,-5,-1,1]),K([3,-1,-8,2,3,-1]),K([-1,0,4,-3,-2,1]),K([-347,211,796,-397,-309,130]),K([-2603,1599,6089,-3009,-2356,980])])
gp: E = ellinit([Polrev([1,5,0,-5,-1,1]),Polrev([3,-1,-8,2,3,-1]),Polrev([-1,0,4,-3,-2,1]),Polrev([-347,211,796,-397,-309,130]),Polrev([-2603,1599,6089,-3009,-2356,980])], K);
magma: E := EllipticCurve([K![1,5,0,-5,-1,1],K![3,-1,-8,2,3,-1],K![-1,0,4,-3,-2,1],K![-347,211,796,-397,-309,130],K![-2603,1599,6089,-3009,-2356,980]]);
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: | \((960a^5-5343a^4+3831a^3+11965a^2-8450a-2262)\) | = | \((3a^5-7a^4-9a^3+17a^2+3a-5)^{15}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 8629188747598184440949 \) | = | \(29^{15}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{49517926748131685406029400211528}{8629188747598184440949} a^{5} - \frac{118018847388936022519470695147732}{8629188747598184440949} a^{4} - \frac{152677702645431525968546689515834}{8629188747598184440949} a^{3} + \frac{305537605996119022369549489960117}{8629188747598184440949} a^{2} + \frac{81336262405475238261621776760232}{8629188747598184440949} a - \frac{129867942024508507713764721191699}{8629188747598184440949} \) | ||
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: | \(0\) |
| 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 1.4813316253083485410290468295415282899 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.40442 \) | ||
| Analytic order of Ш: | \( 625 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3a^5-7a^4-9a^3+17a^2+3a-5)\) | \(29\) | \(1\) | \(I_{15}\) | Non-split multiplicative | \(1\) | \(1\) | \(15\) | \(15\) |
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 |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
29.2-a
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.