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([0,0,0,0]),K([-5,-6,0,1]),K([-3,0,1,0]),K([27,30,-2,-6]),K([7,6,-4,-3])])
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([-5,-6,0,1]),Polrev([-3,0,1,0]),Polrev([27,30,-2,-6]),Polrev([7,6,-4,-3])], K);
magma: E := EllipticCurve([K![0,0,0,0],K![-5,-6,0,1],K![-3,0,1,0],K![27,30,-2,-6],K![7,6,-4,-3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^3-9a-2)\) | = | \((-a^3+a^2+4a+1)^{2}\cdot(2a^2-a-6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 475 \) | = | \(5^{2}\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-15a^3-30a^2+110a+170)\) | = | \((-a^3+a^2+4a+1)^{6}\cdot(2a^2-a-6)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 5640625 \) | = | \(5^{6}\cdot19^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{97722368}{361} a^{3} - \frac{362807296}{361} a^{2} + \frac{32821248}{361} a + \frac{630816768}{361} \) | ||
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 | $\left(a^{2} + a - 1 : a^{3} + a^{2} - 3 a - 3 : 1\right)$ |
| Height | \(0.12510967208128292808850156071241600457\) |
| 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: | \( 0.12510967208128292808850156071241600457 \) | ||
| Period: | \( 316.77972742222254348346266893693895828 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 4.42885354073264 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a+1)\) | \(5\) | \(1\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
| \((2a^2-a-6)\) | \(19\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 475.2-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.