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([4,0,-5,0,1,0]),K([-7,0,6,0,-1,0]),K([-2,0,1,0,0,0]),K([9,0,-8,0,-9,0]),K([-81,0,246,0,-166,0])])
gp: E = ellinit([Polrev([4,0,-5,0,1,0]),Polrev([-7,0,6,0,-1,0]),Polrev([-2,0,1,0,0,0]),Polrev([9,0,-8,0,-9,0]),Polrev([-81,0,246,0,-166,0])], K);
magma: E := EllipticCurve([K![4,0,-5,0,1,0],K![-7,0,6,0,-1,0],K![-2,0,1,0,0,0],K![9,0,-8,0,-9,0],K![-81,0,246,0,-166,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4-a^3+3a^2+2a-1)\) | = | \((-a^4-a^3+3a^2+2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2097152)\) | = | \((-a^4-a^3+3a^2+2a-1)^{42}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 85070591730234615865843651857942052864 \) | = | \(8^{42}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1159088625}{2097152} \) | ||
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 | $\left(\frac{8}{3} a^{5} + \frac{1}{3} a^{4} - 8 a^{3} + 7 a^{2} + 2 : \frac{8}{3} a^{5} + \frac{118}{3} a^{4} + \frac{32}{3} a^{3} - 109 a^{2} + 8 a + 30 : 1\right)$ |
| Height | \(1.0267244893349748108615965951594791037\) |
| 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: | $\left(\frac{1}{3} a^{4} + \frac{13}{3} a^{2} + 2 : 18 a^{4} - \frac{143}{3} a^{2} + \frac{34}{3} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 1.0267244893349748108615965951594791037 \) | ||
| Period: | \( 1.2717055437381528591011406647598375114 \) | ||
| Tamagawa product: | \( 42 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 1.59609 \) | ||
| Analytic order of Ш: | \( 49 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4-a^3+3a^2+2a-1)\) | \(8\) | \(42\) | \(I_{42}\) | Split multiplicative | \(-1\) | \(1\) | \(42\) | \(42\) |
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 |
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 7 and 21.
Its isogeny class
8.1-b
consists of curves linked by isogenies of
degrees dividing 21.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 5 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 162.c2 |
| \(\Q\) | 1296.f2 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-1458.1-p3 |
| \(\Q(\zeta_{9})^+\) | a curve with conductor norm 36864 (not in the database) |
| \(\Q(\zeta_{9})^+\) | 3.3.81.1-8.1-a2 |