Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-1,-4,0,1,0]),K([-2,3,7,-4,-2,1]),K([4,-2,-7,4,2,-1]),K([-74,45,-83,286,20,-98]),K([5435,-28214,-926,52563,-1644,-14247])])
gp: E = ellinit([Polrev([3,-1,-4,0,1,0]),Polrev([-2,3,7,-4,-2,1]),Polrev([4,-2,-7,4,2,-1]),Polrev([-74,45,-83,286,20,-98]),Polrev([5435,-28214,-926,52563,-1644,-14247])], K);
magma: E := EllipticCurve([K![3,-1,-4,0,1,0],K![-2,3,7,-4,-2,1],K![4,-2,-7,4,2,-1],K![-74,45,-83,286,20,-98],K![5435,-28214,-926,52563,-1644,-14247]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 64 \) | = | \(64\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-128)\) | = | \((2)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -4398046511104 \) | = | \(-64^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{661423578612904638319}{64} a^{5} + \frac{2517381707375973330273}{64} a^{4} - \frac{950355314310097681067}{32} a^{3} - \frac{3717389947572253097155}{128} a^{2} + \frac{4067931975116471451485}{128} a - \frac{366235659528699810345}{64} \) | ||
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 | $\left(7 a^{5} + 7 a^{4} - 11 a^{3} - 22 a^{2} - 21 a + 27 : 276 a^{5} - 86 a^{4} - 1042 a^{3} + 402 a^{2} + 542 a - 251 : 1\right)$ |
| Height | \(1.4230937512180818503724189529310385195\) |
| 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: | \( 1.4230937512180818503724189529310385195 \) | ||
| Period: | \( 0.60720808826580085585701469292383259518 \) | ||
| Tamagawa product: | \( 7 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.55323 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(64\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
64.1-b
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.