Base field \(\Q(\zeta_{36})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - 6 x^{4} + 9 x^{2} - 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 9, 0, -6, 0, 1]))
gp: K = nfinit(Polrev([-3, 0, 9, 0, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 9, 0, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,-4,0,1]),K([3,-6,-4,6,1,-1]),K([-1,-3,1,1,0,0]),K([-15,-43,6,43,0,-8]),K([54,87,-40,-80,7,15])])
gp: E = ellinit([Polrev([0,1,0,-4,0,1]),Polrev([3,-6,-4,6,1,-1]),Polrev([-1,-3,1,1,0,0]),Polrev([-15,-43,6,43,0,-8]),Polrev([54,87,-40,-80,7,15])], K);
magma: E := EllipticCurve([K![0,1,0,-4,0,1],K![3,-6,-4,6,1,-1],K![-1,-3,1,1,0,0],K![-15,-43,6,43,0,-8],K![54,87,-40,-80,7,15]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+3a)\) | = | \((a^5-5a^3+4a)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3a^3-9a)\) | = | \((a^5-5a^3+4a)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -19683 \) | = | \(-3^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -140172878133 a^{5} - 180202777029 a^{4} + 609373044090 a^{3} + 783394877511 a^{2} - 254442862521 a - 327105431919 \) | ||
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: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - 2 a^{3} - 2 a^{2} + 2 a + 2 : \frac{1}{3} a^{5} + 2 a^{4} - \frac{8}{3} a^{3} - 10 a^{2} + 5 a + 8 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 17366.840569793901029224063427096624151 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 1.71926 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-5a^3+4a)\) | \(3\) | \(1\) | \(IV^{*}\) | Additive | \(1\) | \(3\) | \(9\) | \(0\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-g
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.