Base field 4.4.5125.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
gp: K = nfinit(Polrev([11, 7, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,-4,0,1]),K([-1,-1,0,0]),K([1,-4,-1,1]),K([-43,393,32,-64]),K([3203,-169,-491,43])])
gp: E = ellinit([Polrev([-4,-4,0,1]),Polrev([-1,-1,0,0]),Polrev([1,-4,-1,1]),Polrev([-43,393,32,-64]),Polrev([3203,-169,-491,43])], K);
magma: E := EllipticCurve([K![-4,-4,0,1],K![-1,-1,0,0],K![1,-4,-1,1],K![-43,393,32,-64],K![3203,-169,-491,43]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-4a-4)\) | = | \((a^3-4a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((172a^3-226a^2-659a+613)\) | = | \((a^3-4a-4)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -17249876309 \) | = | \(-29^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{3402532667241798182971657}{17249876309} a^{3} - \frac{12539596288688205884294718}{17249876309} a^{2} + \frac{1032346455977317685746157}{17249876309} a + \frac{21764174988792336404044948}{17249876309} \) | ||
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: | \( 125.22602337623073048546500778880718661 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.74923297089418 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a-4)\) | \(29\) | \(1\) | \(I_{7}\) | Non-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.6.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
29.1-a
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.