Base field 6.6.1387029.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-6,-6,5,2,-1]),K([4,-3,-18,9,7,-3]),K([4,-6,-14,7,5,-2]),K([-129,37,287,-89,-93,34]),K([-674,245,1502,-535,-476,181])])
gp: E = ellinit([Polrev([3,-6,-6,5,2,-1]),Polrev([4,-3,-18,9,7,-3]),Polrev([4,-6,-14,7,5,-2]),Polrev([-129,37,287,-89,-93,34]),Polrev([-674,245,1502,-535,-476,181])], K);
magma: E := EllipticCurve([K![3,-6,-6,5,2,-1],K![4,-3,-18,9,7,-3],K![4,-6,-14,7,5,-2],K![-129,37,287,-89,-93,34],K![-674,245,1502,-535,-476,181]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-3a^4-3a^3+8a^2+2a-2)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{3}\) |
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: | \((3a^5-7a^4-12a^3+22a^2+10a-2)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{9}\) |
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: | \( 12939256412099098 a^{5} - 16309401572623020 a^{4} - 54249386561334944 a^{3} + 22084272647214278 a^{2} + 25477242594839873 a - 7438316200111393 \) | ||
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 | $\left(-\frac{706}{27} a^{5} + \frac{1925}{27} a^{4} + \frac{1811}{27} a^{3} - \frac{1847}{9} a^{2} - \frac{778}{27} a + 92 : \frac{54521}{243} a^{5} - \frac{147485}{243} a^{4} - \frac{157316}{243} a^{3} + \frac{460319}{243} a^{2} + \frac{74531}{243} a - \frac{204457}{243} : 1\right)$ |
| Height | \(3.2676628565411468851030084605482013398\) |
| 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: | \( 3.2676628565411468851030084605482013398 \) | ||
| Period: | \( 60.291080215039978015744251948330543060 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.01106 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-3a^4-2a^3+8a^2+a-2)\) | \(3\) | \(3\) | \(IV^{*}\) | 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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-a
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.