Base field 3.3.985.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,0,1]),K([4,0,-1]),K([-4,-1,1]),K([-26,-4,4]),K([5,1,-1])])
gp: E = ellinit([Polrev([-4,0,1]),Polrev([4,0,-1]),Polrev([-4,-1,1]),Polrev([-26,-4,4]),Polrev([5,1,-1])], K);
magma: E := EllipticCurve([K![-4,0,1],K![4,0,-1],K![-4,-1,1],K![-26,-4,4],K![5,1,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2-7)\) | = | \((a^2-7)\) |
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: | \((2a^2-6a-7)\) | = | \((a^2-7)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( -841 \) | = | \(-29^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{18499880885}{841} a^{2} - \frac{15485040525}{841} a - \frac{113522068011}{841} \) | ||
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 | $\left(-\frac{2158}{1445} a^{2} + \frac{1609}{1445} a + \frac{15137}{1445} : \frac{350559}{122825} a^{2} - \frac{49333}{24565} a - \frac{2484434}{122825} : 1\right)$ |
Height | \(3.5532727336487633413398422980386711825\) |
Torsion structure: | \(\Z/5\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(1 : -a^{2} + a + 4 : 1\right)$ |
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: | \( 3.5532727336487633413398422980386711825 \) | ||
Period: | \( 128.40433499784394463817444935141890408 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(5\) | ||
Leading coefficient: | \( 3.48900310 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-7)\) | \(29\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
29.3-c
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.