Base field 5.5.89417.1
Generator \(a\), with minimal polynomial \( x^{5} - 6 x^{3} - x^{2} + 8 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 8, -1, -6, 0, 1]))
gp: K = nfinit(Polrev([3, 8, -1, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 8, -1, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,0,-4,0,1]),K([-4,-7,4,2,-1]),K([0,1,0,0,0]),K([-458,-923,610,574,-291]),K([10852,23224,-15498,-13684,7024])])
gp: E = ellinit([Polrev([2,0,-4,0,1]),Polrev([-4,-7,4,2,-1]),Polrev([0,1,0,0,0]),Polrev([-458,-923,610,574,-291]),Polrev([10852,23224,-15498,-13684,7024])], K);
magma: E := EllipticCurve([K![2,0,-4,0,1],K![-4,-7,4,2,-1],K![0,1,0,0,0],K![-458,-923,610,574,-291],K![10852,23224,-15498,-13684,7024]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-3a+2)\) | = | \((a^3-a^2-3a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2a^4+4a^3+7a^2-16a-5)\) | = | \((a^3-a^2-3a+2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -14641 \) | = | \(-11^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{64801139993982903601201}{14641} a^{4} + \frac{79742135469152888974129}{14641} a^{3} + \frac{290678984684285122184960}{14641} a^{2} - \frac{292899070194533892815349}{14641} a - \frac{157977896987517584455026}{14641} \) | ||
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 | $\left(-a + 3 : 17 a^{4} - 35 a^{3} - 33 a^{2} + 56 a + 23 : 1\right)$ |
| Height | \(0.20986498322965686577225309587572416071\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(3 a^{4} - \frac{25}{4} a^{3} - \frac{25}{4} a^{2} + \frac{19}{2} a + \frac{31}{4} : -\frac{7}{2} a^{4} + \frac{39}{8} a^{3} + \frac{91}{8} a^{2} - \frac{83}{8} a - \frac{31}{4} : 1\right)$ |
| 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: | \( 0.20986498322965686577225309587572416071 \) | ||
| Period: | \( 1457.3370136182447877900487661030786913 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.55699535 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-3a+2)\) | \(11\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
11.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.