Base field 5.5.161121.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 3, -6, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 3, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 3, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-6,-10,11,3,-2]),K([-2,-6,0,1,0]),K([0,-5,0,1,0]),K([1790,6565,-5889,-1723,1135]),K([47655,157332,-139745,-41489,26992])])
gp: E = ellinit([Polrev([-6,-10,11,3,-2]),Polrev([-2,-6,0,1,0]),Polrev([0,-5,0,1,0]),Polrev([1790,6565,-5889,-1723,1135]),Polrev([47655,157332,-139745,-41489,26992])], K);
magma: E := EllipticCurve([K![-6,-10,11,3,-2],K![-2,-6,0,1,0],K![0,-5,0,1,0],K![1790,6565,-5889,-1723,1135],K![47655,157332,-139745,-41489,26992]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-a^3-6a^2+3a+4)\) | = | \((a^4-a^3-6a^2+3a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 3 \) | = | \(3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^2-a-2)\) | = | \((a^4-a^3-6a^2+3a+4)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -27 \) | = | \(-3^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{10947981839295713687464790827586754793436}{27} a^{4} - \frac{16910116865672541902732883385121893663963}{27} a^{3} - \frac{56478851274393237585283385062065675883994}{27} a^{2} + \frac{63601626922138511052720584288730205307096}{27} a + \frac{20103252580374097231986758925581632058597}{27} \) | ||
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: | \( 0.038247934103890984998599933095103726834 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.686349502 \) | ||
| Analytic order of Ш: | \( 2401 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-a^3-6a^2+3a+4)\) | \(3\) | \(3\) | \(I_{3}\) | 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 |
| \(7\) | 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
3.1-b
consists of curves linked by isogenies of
degrees dividing 21.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.